Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media

Transcription

1 University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media Cody Samuel Lorton University of Tennessee - Knoxville, clorton@utk.edu Recommended Citation Lorton, Cody Samuel, "Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media. " PhD diss., University of Tennessee, This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.edu.

2 To the Graduate Council: I am submitting herewith a dissertation written by Cody Samuel Lorton entitled "Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. We have read this dissertation and recommend its acceptance: Michael W. Berry, Ohannes A. Karakashian, Steven M. Wise Original signatures are on file with official student records. Xiaobing H. Feng, Major Professor Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School

3 University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media Cody Samuel Lorton This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact

4 To the Graduate Council: I am submitting herewith a dissertation written by Cody Samuel Lorton entitled "Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. We have read this dissertation and recommend its acceptance: Michael W. Berry, Ohannes A. Karakashian, Steven M. Wise Original signatures are on file with official student records. Xiaobing H. Feng, Major Professor Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School

5 Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Cody Samuel Lorton August 014

6 c by Cody Samuel Lorton, 014 All Rights Reserved. ii

7 This dissertation is dedicated to Jesus Christ my Lord and Savior, my wife Rebekah, and my children Bella and Shepherd. iii

8 Acknowledgements I would first like to thank my advisor, Professor Xiaobing Feng, for his excellent leadership and instruction throughout my graduate studies. I could not ask for a better advisor and I know that his insight, guidance, and hard work have provided a lasting foundation for my research career. Secondly, I would like to thank the remaining members of my dissertation committee Professors Steven Wise, Ohannes Karakashian, and Michael Berry for their time and guidance. I would also like to thank all those in the mathematics department at the University of Tennessee that helped me along during my graduate studies. In particular, I must thank Professors Steven Wise, Ohannes Karakashian, and Suzanne Lenhart for helping lay the foundation of my career in applied mathematics. I must also thank Pam Armentrout and Ben Walker for their support as well. I would like to thank my wife Rebekah and my children Bella and Shepherd. I could not accomplish all that I have without their support and love. Lastly, I would like to thank God. It is only by God s loving will that I completed this dissertation. iv

9 Abstract This dissertation consists of four integral parts with a unified objective of developing efficient numerical methods for high frequency time-harmonic wave equations defined on both homogeneous and random media. The first part investigates the generalized weak coercivity of the acoustic Helmholtz, elastic Helmholtz, and time-harmonic Maxwell wave operators. We prove that such a weak coercivity holds for these wave operators on a class of more general domains called generalized star-shape domains. As a by-product, solution estimates for the corresponding Helmholtz-type problems are obtained. The second part of the dissertation develops an absolutely stable i.e. stable in all mesh regimes interior penalty discontinuous Galerkin IP-DG method for the elastic Helmholtz equations. A special mesh-dependent sesquilinear form is proposed and is shown to be weakly coercive in all mesh regimes. We prove that the proposed IP-DG method converges with optimal rate with respect to the mesh size. Numerical experiments are carried out to demonstrate the theoretical results and compare this method to the standard finite element method. The third part of the dissertation develops a Monte Carlo interior penalty discontinuous Galerkin MCIP-DG method for the acoustic Helmholtz equation defined on weakly random media. We prove that the solution to the random Helmholtz problem has a multi-modes expansion i.e., a power series in a mediumrelated small parameter. Using this multi-modes expansion an efficient and accurate numerical method for computing moments of the solution to the random Helmholtz v

10 problem is proposed. The proposed method is also shown to converge optimally. Numerical experiments are carried out to compare the new multi-modes MCIP-DG method to a classical Monte Carlo method. The last part of the dissertation develops a theoretical framework for Schwarz preconditioning methods for general nonsymmetric and indefinite variational problems which are discretized by Galerkin-type discretization methods. Such a framework has been missing in the literature and is of great theoretical and practical importance for solving convection-diffusion equations and Helmholtz-type wave equations. Condition number estimates for the additive and hybrid Schwarz preconditioners are established under some structure assumptions. Numerical experiments are carried out to test the new framework. vi

11 Table of Contents 1 Introduction The State of the Art Summary of this Dissertation Notation Generalized Weak Coercivity of Reduced Wave Operators 9.1 Introduction to Generalized Weak Coercivity and Generalized Star- Shape Domains The Scalar Helmholtz Operator The Elastic Helmholtz Operator The Time-Harmonic Maxwell Operator Applications to Stability Estimates Absolutely Stable Discontinuous Galerkin Methods for the Elastic Helmholtz Equations Formulation of the IP-DG Method Some Properties of the IP-DG Method Asymptotic Error Estimates Elliptic Projection and its Error Estimates Asymptotic Error Estimates Via Schatz Argument Pre-asymptotic Error Estimates Stability Estimates in the Pre-Asymptotic Mesh Regime vii

12 3.3. Error Estimates for the IP-DG Method Numerical Experiments Stability Error IP-DG vs. FEM A Multi-modes Monte Carlo Interior Penalty Discontinuous Galerkin Method for Acoustic Wave Scattering in Random Media Introduction PDE Analysis Preliminaries Wave-number Explicit Solution Estimates Multi-modes Representation of the Solution and its Finite Modes Approximations Monte Carlo Discontinuous Galerkin Approximation of the Truncated Multi-modes Expansion UN ε DG Notations IP-DG Method for Deterministic Helmholtz Problem MCIP-DG Method for Approximating EU ε n The Overall Numerical Procedure The Numerical Algorithm, Linear Solver and Computational Complexity Convergence Analysis Numerical Experiments MCIP-DG with Multi-modes Expansion Compared to Classical MCIP-DG More Numerical Tests Schwarz Space Decomposition Methods for Nonsymmetric and Indefinite Problems 140 viii

13 5.1 Introduction Functional Setting and Statement of Problems Variational Problem Discrete Problem Main Objective An Abstract Schwarz Framework for Nonsymmetric and Indefinite Problems Main Assumptions and Main Idea Space Decomposition and Local Solvers Additive Schwarz Method Multiplicative Schwarz Method A Hybrid Schwarz Method An Abstract Schwarz Preconditioner Theory for Nonsymmetric and Indefinite Problems Structure Assumptions Condition Number Estimate for P ad Condition Number Estimate for P hy Application to DG Discretizations for Convection-diffusion Problems Discontinuous Galerkin Approximations Partial Analysis of the 1-D Convection Diffusion Problem Numerical Experiments Future Directions 197 Bibliography 199 Vita 10 ix

14 List of Tables 4.1 CPU times required to compute the multi-modes MCIP-DG approximation Ψ h N and the classical MCIP-DG approximation Ψ h Relative error in the L -norm between the multi-modes MCIP-DG approximation Ψ h 3 and the classical MCIP-DG approximation Ψ h Relative error in the L -norm between the multi-modes MCIP-DG approximation Ψ h N and the classical MCIP-DG approximation Ψ h Performance of three Schwarz methods on Test Performance of three Schwarz methods on Test Performance of three Schwarz methods on Test Performance of three Schwarz methods on Test x

15 List of Figures.1 An example of a domain Ω of interest Example of the triangulation T 1/ Plot of Re u h for ω = 50 and h = 1/70. Both a top down view left and a side view right are shown Plot of Re u h for ω = 100 and h = 1/10. Both a top down view left and a side view right are shown Plots of u h 1,h and u F EM h 1,h Log-log plot of the relative error for the IP-DG approximation measured in the H 1 -seminorm for different values of ω Relative error of the IP-DG approximation measured in the H 1 seminorm computed for different values of ω and h is chosen to satisfy the given constraints The left plot is of Reu h solid red line vs. Reu dashed blue line for h = 1/50. The right plot is of Reu F h EM solid red line vs. Reu dashed blue line for h = 1/ The left plot is of Reu h solid red line vs. Reu dashed blue line for h = 1/10. The right plot is of Reu F h EM solid red line vs. Reu dashed blue line for h = 1/ The left plot is of Reu h solid red line vs. Reu dashed blue line for h = 1/00. The right plot is of Reu F h EM solid red line vs. Reu dashed blue line for h = 1/ xi

16 4.1 Triangulation T 1/ Discrete average media 1 M M j=1 αω j, left and a sample media αω, right computed for h = 1/0, ε = 0.1, η, x U[ 1, 1], and M = left Relative error in the L -norm between Ψ h N computed using the multi-modes MCIP-DG method and Ψ h computed using the classical MCIP-DG method. right ε N vs. N for N = 1,,, Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.0, η, x U[ 1, 1], and M = Cross sections of Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.0, η, x U[ 1, 1], and M = 1000, over the line y = x Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.1, η, x U[ 1, 1], and M = Cross sections of Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.1, η, x U[ 1, 1], and M = Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.5, η, x U[ 1, 1], and M = Cross sections of Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.5, η, x U[ 1, 1], and M = Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.8, η, x U[ 1, 1], and M = Cross sections of Re Ψ3 h left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.8, η, x U[ 1, 1], and M = Dependence of κ A P ad and κ A P hy on J in Test Spectrum plots from Test Spectrum plots from Test xii

17 Chapter 1 Introduction As a fundamental mechanism for energy transmission, wave phenomena are ubiquitous in our world. Waves are determined by their sources and the media in which they propagate. Wave scattering describes the physical phenomena in which wave propagation is changed due to some non-uniformity in the medium in which the wave is traveling. Wave scattering problems have applications in many scientific fields including communications, defense, aviation, geoscience, medical science, manufacturing, etc. The goal of this dissertation is to develop efficient numerical methods for high frequency time-harmonic wave equations defined on both homogeneous and random media. Specifically, it focuses on three basic mathematical models of wave scattering and propagation. These are the acoustic Helmholtz, elastic Helmholtz, and timeharmonic Maxwell s equations. The first wave scattering problem we will consider is the acoustic/scalar Helmholtz problem given by u k u = f in Ω, 1.1 u n + + iku = g on Ω +, 1. u = 0 on Ω

18 Here, Ω R d d = 1,, 3 is a domain that consists of some acoustic medium and u : Ω C is the pressure of the medium. k is the wave number, defined by k := ω, c where ω, c > 0 are the angular frequency and speed of the wave in Ω, respectively. f is the external source. Ω is decomposed into two pieces Ω + and Ω. n +, n denotes the unit outward normal vectors on Ω + and Ω, respectively. Typically, wave propagation problems are posed on large or unbounded domains complemented with a far-field radiation condition. For computational purposes, we choose to utilize a truncated domain. Ω + represents the boundary from this truncation. When g = 0, 1. is a first order absorbing boundary condition [35], which is an artificial boundary condition that absorbs incoming waves at the boundary. Ω is the scattering portion of the domain boundary. 1.3 ensures that the scattering boundary Ω is sound soft. The acoustic Helmholtz problem comes from seeking time-harmonic solutions or applying Fourier transforms in t to the well-known acoustic wave problem 1 c U tt U = F 1 c U t + U = G n + U = 0 in Ω 0,, on Ω + 0,, on Ω 0,, U = U t = 0 in Ω {t = 0}. Here u, f, g from take the form ux = fx = gx = e iωt Ux, tdt, e iωt F x, tdt, e iωt Gx, tdt.

19 Computing solutions to is known as the frequency domain treatment for wave problems [9, 30]. This approach is favorable, because for a set of chosen frequencies one can compute time-harmonic solutions in parallel by solving a set of independent acoustic Helmholtz problems. Also, the use of frequency specific timeharmonic waves often arise from many applications. The second problem that we will consider is the elastic Helmholtz problem given by ω ρu div σu = f in Ω, 1.4 iωau + σun = g on Ω. 1.5 Similar to the acoustic Helmholtz problem, arise from seeking timeharmonic solutions to the well-known linear elastic wave equations. Ω R d = 1,, 3 is a domain that consists of some elastic medium and u : Ω C d is the displacement vector of that medium. ω, ρ are the angular frequency of the elastic wave and the density of the elastic medium, respectively. For the elastic Helmholtz equation, the wave number is given by k = ρω. σu denotes the stress tensor defined by σu := µεu + λdiv ui, εu := 1 u + u T. Here, µ, λ > 0 are the Lamé constants for the elastic medium Ω and εu is called the strain tensor. We do not consider a scattering portion of the boundary in for simplicity. Similar to 1., when g = 0, 1.5 is a first order absorbing boundary condition [35]. A is a d d symmetric positive-definite constant matrix. Lastly, we consider the time-harmonic Maxwell s equations given by curl curl E k E = f in Ω, 1.6 curl E n iλe T = g on Ω

20 arise from seeking time-harmonic solutions to the well-known Maxwell s equations c.f. [5]. Ω R 3 and E : Ω C 3 is the electrical field of Ω. E T = n E n is the tangential part of E. The wave number k is defined as k = ω µ 0 ε 0, where ω > 0 is the angular frequency of the wave, ε 0 > 0 is the electrical permittivity of the medium, and µ 0 > 0 is the magnetic permeability of the medium. Similar to the elastic Helmholtz problem, we will not consider a scattering portion of the boundary for simplicity. 1.7 is the standard impedance boundary condition, with λ > 0 called the impedance constant. Because the acoustic Helmholtz, elastic Helmholtz, and time-harmonic Maxwell s problems all arise by seeking time-harmonic solutions to wave problems and thus have similar characteristics, these three problems will be referred to as Helmholtztype problems in this dissertation. 1.1 The State of the Art Many numerical methods have been developed for the three Helmholtz-type problems in homogeneous media, i.e. for constant wave number k. These include finite difference FD, finite volume FV, finite element FE, and discontinuous Galerkin DG methods [1,, 3, 6, 7, 10, 13, 19, 0, 3, 6, 9, 30, 3, 36, 38, 48, 51, 5, 54, 53, 59, 60, 61, 64, 67, 70, 75, 76]. This section will discuss some of the challenges that arise from solving the three Helmholtz-type problems numerically. Recall that Helmholtz-type problems are wave problems. Solutions to these problems are oscillatory with wave length l = π/k. Enough grid points must be used in the spacial domain to resolve the wave. The widely accepted rule-of-thumb is to use 6 1 mesh/grid points per wavelength. This rule-of-thumb was proved rigorously for the linear FE method for the 1-D acoustic Helmholtz problem [54, 53]. This yields a mesh constraint of kh = O1, where h is the mesh size. Meshes satisfying this mesh constraint make up the so-called pre-asymptotic mesh regime. Therefore, 4

21 in the high frequency case, discretizing the Helmholtz-type problems yields a large system of linear equations that must be solved. In the case of linear FE method for the 1-D acoustic Helmholtz problem, the authors of [54] showed that the H 1 error bound for the FE solution contains a term of order k 3 h. This term is called the pollution term and an increase in error as one increases the wave number k under the constraint kh = O1 is called the pollution effect. The authors of [13, 9, 54] showed that the pollution effect is inherent in Helmholtz-type problems and also leads to a loss of stability of standard discretization techniques. To eliminate the pollution effect, a more stringent mesh constraint k h = O1, called the asymptotic mesh constraint, is used. It is under this constraint that stability is proved for standard discretization techniques applied to Helmholtz-type problems. Shen and Wang obtained an absolutely stable i.e. stable for all k, h > 0 spectral Galerkin discretization for the radially symmetric acoustic Helmholtz equation in [73]. Feng and Wu obtained absolutely stable interior penalty discontinuous Galerkin IP-DG discretizations for the acoustic Helmholtz and time-harmonic Maxwell s equations in [4, 43, 44]. Feng and Xing obtained an absolutely stable local discontinuous Galerkin LDG method for the acoustic Helmholtz equation in [45]. As noted previously, for k large one must solve a large linear system of equations in order to solve Helmholtz-type problems. From 1.1,1.4, and 1.6 we see that for k or ω large the Helmholtz-type PDE operators are indefinite. Thus, any discretization method applied to Helmholtz-type PDEs yield indefinite, and illconditioned linear systems. It is known that standard iterative methods do not work well when applied to Helmholtz-type problems. In fact, many are not convergent c.f. [37]. There is no framework in place to analyze multi-level solvers/preconditioners, such as multi-grid and Schwarz domain decomposition methods, for indefinite problems like the Helmholtz-type problems. Also, if one must adhere to the stringent mesh constraint k h = O1 in the high frequency case, practical coarse mesh spaces for multi-level solvers cannot be implemented. 5

22 1. Summary of this Dissertation This dissertation contains five additional chapters. In Chapter, we study the three Helmholtz-type problems at the PDE level. In particular, we show that all three Helmholtz-type PDEs satisfy a generalized weak coercivity property. This generalized weak coercivity property was proved to hold for the time-harmonic Maxwell s equations in [43]. The techniques used to prove these generalized weak coercivity properties were first used in [7] and rely on Rellich identities for the Helmholtz-type operators as well as a star-shape condition on the domain Ω. Because a star-shape condition can be viewed as restrictive, the analysis in Chapter is carried out on generalized star-shape domains. As a corollary of the generalized weak coercivity property, solution estimates are proved in energy norms for each Helmholtztype problem. Chapter 3 develops an absolutely stable interior penalty discontinuous Galerkin IP-DG method for the elastic Helmholtz problem. Recall that this was already done for the acoustic Helmholtz and time-harmonic Maxwell s problem [4, 43, 44]. This chapter uses new techniques, introduced in [4, 43, 44], to obtain stability and optimal in h error estimates in the pre-asymptotic mesh regime. Analysis in the asymptotic mesh regime is also carried out using the standard Schatz argument. Numerical experiments are provided to demonstrate the theoretical results presented in this chapter. In Chapter 4, we develop a Monte Carlo interior penalty discontinuous Galerkin MCIP-DG method for the acoustic Helmholtz problem in random media. The random media is characterized by use of a random wave number in the acoustic Helmholtz problem. In this chapter, we show that when this random wave number is a random perturbation of some constant wave number, the solution takes the form of a power series expansion in the perturbation parameter. We call this series expansion the multi-modes expansion. Using this multi-modes expansion, an efficient and accurate MCIP-DG method is obtained. Numerical experiments presented to 6

23 show that the multi-mode MCIP-DG method is accurate compared to the classical MCIP-DG method and much more efficient. There is no general framework to study Schwarz preconditioners for general non-hermitian and indefinite variational problems. This includes Helmholtz-type problems. As a first step to meet this challenge, in Chapter 5, we develop a general framework to analyze Schwarz preconditioners for real-valued non-symmetric and indefinite variational problems. In this chapter the theoretical framework is introduced and different Schwarz preconditioners are developed and analyzed. This new framework is designed as a generalization of the existing Schwarz framework given in [77]. Extensive numerical experiments are also conducted to demonstrate some properties of Schwarz preconditioners applied to a non-symmetric problem. Though this framework does not apply directly to the three Helmholtz-type problems, it is our hope that this initial step will lead to a generalization that also applies to these Helmholtz-type problems. Lastly, Chapter 6 discusses a number of future research directions that come from this dissertation. 1.3 Notation This dissertation adopts many standard notation conventions. Much of the notation is explained when it is introduced, but we define some standard notations here that will be used throughout. H β Ω will be used to denote the standard Sobolev space W β, Ω. For any S Ω and Σ Ω, let, S and, Σ denote the standard L -inner products defined by u, v S := u v dx, u, v Σ := u v ds, S S for all u, v L S and u, v L Σ, respectively. 7

24 A bold-face font will be used to emphasize a vector or vector valued function, such as x R d or u : S C d. With this in mind, we use the following bold-face convention for identifying vector-valued function spaces: { } L p S := v : S C d vi L p S for all i = 1,,, d, { } H β S := v : S C d vi H β S for i = 1,,, d. 8

25 Chapter Generalized Weak Coercivity of Reduced Wave Operators.1 Introduction to Generalized Weak Coercivity and Generalized Star-Shape Domains This section introduces two new concepts; namely, generalized weak coercivity and generalized star-shape domains. As was already discussed, the Helmholtz-type operators are indefinite. Thus, one cannot expect the sesquilinear forms used to define the weak formulation of the Helmholtz-type operators to be coercive. In fact, in the case of Helmholtz-type operators one cannot even expect a weak coercivity property. Instead, for Helmholtz-type operators a generalized weak coercivity property of the form sup v V Im au, v v V takes the place of standard weak coercivity. Re au, v + sup C u E u E.1 v W v W Such a generalized weak coercivity property can be used to obtain a-priori wave-number explicit estimates for solutions of the three Helmholtz-type PDEs. 9

26 Generalized weak coercivity is also valuable in the development of novel discretization methods and linear solvers that are tailored to these Helmholtz-type problems. Specifically, the techniques employed in the proofs of the generalized weak coercivity properties can be useful in the development of absolutely stable discretization methods for Helmholtz-type problems c.f. [4, 43, 44, 50]. That is, methods that are stable regardless of the mesh size h. Such an absolutely stable method for the elastic Helmholtz equation is developed and analyzed in Chapter 3. Absolutely stable methods are necessary to provide practical coarse mesh spaces, a key component for any multi-level method such as multi-grid or multi-level domain decomposition methods. With multi-level methods in mind, the analysis of two-level domain decomposition for non-symmetric and indefinite linear problems in real valued Banach spaces is the focus of Chapter 5. The analysis in this chapter is based on a weak coercivity condition. It is believed that for non-symmetric and indefinite linear problems in complex valued Banach spaces the existing framework can be extended to include problems that satisfy a generalized weak coercivity condition in lieu of the standard weak coercivity condition. This is yet another motivation to study such generalized weak coercivity conditions. The techniques used to obtain generalized weak coercivity properties of the Helmholtz-type operators are adapted from the techniques in [7, 43, 50]. The analysis found in these sources relies on a star-shape condition on the domain Ω R d. That is, for Ω there exists x 0 Ω and a positive constant c = cω such that for α = x x 0 the following condition holds: α n c on Ω. Practically, this constraint on the domain Ω is adequate when a scattering object is not present. In this case Ω is usually a truncation of a large or unbounded domain and can be chosen to meet this requirement. On the other hand, for a scattering 10

27 problem, a condition like this can be restrictive. This is due to the fact that portions of the boundary can be attributed to the scattering object. With this in mind, this chapter is used to establish less restrictive generalized star-shape conditions on Ω for each Helmholtz-type operator. In particular, these new generalized star-shape conditions allow the existing analysis to hold while admitting more exotic geometry. These generalized star-shape domains are designed for each Helmholtz-type operator, separately. This idea does away with the one-size fits all nature of the standard star-shape condition and replaces it with operator friendly domain constraints. This chapter is organized as follows: Sections..4 are used to tailor a generalized star-shape condition for each Helmholtz-type operator and prove a generalized weak coercivity condition for each operator. Section.5 applies the results of the previous sections to obtain stability estimates for each Helmholtz-type problem.. The Scalar Helmholtz Operator First, a generalized star-shape domain for the scalar Helmholtz operator is defined. We consider an acoustic domain Ω = Ω + \ Ω. Here, Ω + is the truncation of some unbounded acoustic medium and Ω Ω + is some scattering object in the medium. For the existing analysis to hold using a classic star-shape condition one requires that Ω + and Ω are both star-shape domains with respect to the same point x 0 Ω. An example of such a domain is given in Figure.1. 11

28 Figure.1: An example of a domain Ω of interest. To generalize this idea, it is required that Ω is a domain such that there exists a vector field α C 1 Ω that satisfies the following conditions: α j = α j x j,. α n + c + > 0 on Ω +,.3 α n c > 0 on Ω,.4 α R in Ω,.5 c 1 div α c in Ω,.6 { } αi min c 3 > 0 in Ω and i = 1,... d,.7 x i c 1 c + c 3 c 4 > 0 in Ω,.8 where Ω = Ω + Ω and n +, n are the outward normal vectors to Ω + and Ω, respectively. A domain Ω that admits a vector field α as described above will be called a generalized star-shape domain for the scalar Helmholtz equation. Remark..1. a This is a true generalization of the concept of a star-shape domain in the sense that any star-shape domain does satisfy the above properties. This includes the case discussed above c.f. Figure.1. 1

29 b The motivation for all of the generalized star-shape conditions introduced in this chapter comes from the use of Rellich identities for these specific Helmholtz-type operators. These identities are key in the techniques used to prove the generalized weak coercivity properties for each Helmholtz-type operator. c It is conjectured that this generalization allows room for interesting computational domains that are not star-shape domains in the classical sense. At this point, no such examples are known, and this will be an item explored in future research. For the rest of this section, Ω is assumed to be a generalized star-shape domain for the scalar Helmholtz equation. Recall that the generalized weak coercivity property is a property of the weak form of Helmholtz-type PDEs. Therefore, the weak form of will need to be given. For the sake of completeness, the weak form is derived in the preceding lines. Begin by multiplying 1.1 by v C Ω and integrating over all Ω. To this, integration by parts and 1. are applied. These steps yield the following sequence of identities: u, v Ω k u, v Ω = f, v Ω, u u, v Ω n, v k u, v Ω = f, v Ω, Ω u u u, v Ω, v, v k u, v Ω = f, v Ω, n + Ω + n Ω u u, v Ω + ik u, v Ω+, v k u, v Ω = f, v Ω + g, v Ω+. n Ω From the above identity, we observe that an appropriate solution space for the weak formulation of is given by { } V := u H 1 Ω u = 0 on Ω and u L Ω. 13

30 Now the weak form of is defined in the following way: Find u V such that au, v = f, v Ω + g, v Ω+ v H 1 Ω,.9 where a, is a sesquilinear form defined on V H 1 Ω given by au, v := u, v Ω k u, v Ω + ik u, v Ω+ u, v..10 n Ω The goal of this subsection is to prove a generalized weak coercivity condition c.f..1 for the above sesquilinear form a,. To accomplish this goal we rely on the following Rellich identities quoted from [7]: Lemma... Let u H Ω and α C 1 Ω. Then the following identity holds: Reu, u α Ω = 1 div α, u Ω 1 α n+, u Ω. Lemma..3. Let u H Ω and α C 1 Ω. Then the following identity holds: Re u, u α Ω = 1 div α, u + 1 Ω α n+, u Ω d d u +, α j u. x i x i x j Ω i=1 j=1 With these Rellich identities in hand, the following generalized weak coercivity property for the scalar Helmholtz operator is obtained: Theorem..4. Let Ω R d be a generalized star-shape domain with α C 1 Ω satisfying..8. Then for any u V the following generalized weak coercivity 14

31 property holds for the sesquilinear form a, : sup v V Im au, v Re au, v + sup v E v H 1 Ω v L Ω 1 γ u E, where { [c γ := max 4c 3 + c k + 1R ] } 1, M, M := kr + kr + c +, c + k u L Ω := k c 4 u L Ω + c + u L Ω +, u E := k c 4 u L Ω + c 4 u L Ω + c + u L Ω + + c + u L Ω + + c u L Ω 1. 1 Proof. In this proof, we assume that u H Ω V. This is possible because u V can be approximated by a sequence of smooth functions that converge to u in E. Once the result is obtained for u H Ω a limit can be applied to obtain the result for u V. For the sake of brevity, these details are suppressed in the steps to follow. Begin by setting v = u in.10 and taking the real and imaginary part separately. This yields the following identities: Re au, u = u L Ω k u L Ω,.11 Im au, u = k u L Ω +..1 As will be a common theme for the analysis of all three Helmholtz-type problems, the indefiniteness of the scalar Helmholtz operator shows up here in an adverse way. That is, the signs of the terms on the right hand side of.11 are different. Thus the use of this one test function is not sufficient. For this reason, we employ a second test function v = u α, motivated by the above Rellich identities. Using this test 15

32 function in.10 yields u Re au, v = Re u, v Ω k Reu, v Ω k Im u, v Ω+ Re, v. n Ω.13 We substitute the Rellich identities from Lemma.. and Lemma..3 into.13 and rearrange the terms to get k div α, u 1 Ω div α, u 3 + α i, u i Ω x i x i = k α n+, u Ω + 1 α n+, u Ω + 1 α n, u Ω u + Re, v + k Im u, v Ω+ + Re au, v n Ω = k α n+, u Ω + 1 α n+, u Ω α n, u Ω + k Im u, v Ω+ + Re au, v. i=1 Ω Notice that the first line above uses. and we get the last equality because u = u n n on Ω since u = 0 on Ω. Using the conditions on α that are found in.3.6 and multiplying the previous inequality through by produces the following inequality: k c 1 u L Ω c u L Ω + c 3 u L Ω k α n +, u Ω+ c + u L Ω c u L Ω + k Im u, v Ω+ + Re au, v. 16

33 Adding c c 3 times.11 and c + k times.1 to the above inequality yields k c 1 c + c 3 u L Ω + c + u L Ω + k α n +, u Ω+ c + u L Ω c u L Ω + k Im u, v Ω+ + Re a u, v + c c 3 u + c + k Im au, u. Applying.8 and adding c 4 times.11 gives k c 4 u L Ω + c 4 u L Ω + c + u L Ω + k α n +, u Ω+ c + u L Ω c u L Ω + k Im u, v Ω+ + Re a u, v + c c 3 + c 4 u + c + Im au, u. k At this point, we apply Cauchy-Schwarz and Young s inequalities in conjunction with.1 to the previous inequality to obtain the following: k c 4 u L Ω + c 4 u L Ω + c + u L Ω + k R u L Ω + c + u L Ω + c u L Ω + c + Im au, u k + Re a u, v + c c 3 + c 4 u + kr u L Ω + u L Ω + k R u L Ω + c + u L Ω + c u L Ω + c + Im au, u k + Re a u, v + u + k R c c 3 + c 4 = c + u L Ω + c u L Ω + + Re a u, v + c c 3 + c 4 u. Consequently, u L c Ω + + c + + u L Ω + kr + kr + c + Im au, u c + k u E M Im au, u + Re a u, 4v c 4c 3 + c 4 u

34 Let ˆv = 4v c 4c 3 + c 4 u. Putting this test function into L Ω yields ] ˆv L Ω c 4c 3 + c 4 [k c 4 u L Ω + c + u L Ω + ] + 16R [k c 4 u L Ω + c + u L Ω + [ c 4c 3 + c k + 1R ] u E. Finally, this inequality along with.14 implies that sup v V Im au, v Re au, v + sup v E v H 1 Ω v L Ω 1 γ Im au, u u E + Im au, u u E + Re au, ˆv ˆv L Ω Re au, ˆv [c 4c 3 + c k + 1R ] 1 u E M Im au, u + Re au, ˆv u E 1 γ u E. Hence the generalized weak coercivity condition holds..3 The Elastic Helmholtz Operator In this section, the focus is turned to the elastic Helmholtz operator. This operator is similar to the scalar Helmholtz operator. Due to this similarity, the analysis for the elastic Helmholtz operator should follow that of the scalar Helmholtz operator. This section is restricted to the case in which Ω = and thus, Ω = Ω +, where Ω is the elastic medium. Such a restriction is made to compensate for the added difficulty in working with vector-valued functions. Ω is defined to be a generalized star-shape 18

35 domain for which there exists α C 1 Ω such that the following properties hold: α j = α j x j,.15 α n c + > 0 on Ω,.16 α R in Ω,.17 α i x i = c 1 > 0 in Ω and i = 1,... d..18 As was the case in Section., the analysis of this section relies on Rellich identities for the elastic Helmholtz operator. These Rellich identities are the reason behind the constraints placed on the domain. Unfortunately, the Rellich identities for the elastic Helmholtz operator do not yield as much as those for the scalar Helmholtz operator. This is mainly a result of the increase in complexity when moving from scalar-valued functions to vector-valued functions. For this reason the generalized star-shape domain criterion for the elastic Helmholtz operator is more restrictive than that of the scalar Helmholtz operator. A less restrictive domain might be possible, but different techniques will be needed to attain a generalized weak coercivity condition. Now with a generalized star-shape domain defined for the elastic Helmholtz operator, a weak formulation of will be derived. To begin, multiply 1.4 with a smooth function v C Ω and integrate over Ω to obtain ω ρu, v Ω div σu, v Ω = f, v Ω..19 Since σu is symmetric the following product rule for the divergence holds: div σuv = div σuv + σu : v. 19

36 This identity together with the divergence theorem yields div σu, v Ω = div σuvdx + σu, v Ω Ω = σun, v Ω + σu, v Ω..0 From Lemma 3 of [7] one obtains the following useful identity: σu : v = λdiv udiv v + µεu : εv. With this identity.0 becomes div σu, v Ω = λdiv u, div v Ω + µεu, εv Ω σun, v Ω..1 Applying this integration by parts formula along with 1.5 to.19 gives λdiv u, div v Ω + µεu, εv Ω ω ρu, v Ω + iω Au, v Ω = f, v Ω + g, v Ω. Thus, a weak formulation of the elastic Helmholtz equations is given by: find u H 1 Ω such that au, v = f, v Ω + g, v Ω v H 1 Ω,. where a, is defined on H 1 Ω H 1 Ω by au, v := λdiv u, div v Ω + µεu, εv Ω ω ρu, v Ω + iω Au, v Ω..3 Now that the weak formulation is established on a generalized star-shape domain, the focus of this section shifts to obtaining a generalized weak coercivity property for a,. As stated previously, this will require the use of some Rellich identities for the 0

37 elastic Helmholtz operator. These Rellich identities were established in [7] and are quoted below as the following two lemmas. Lemma.3.1. For u H Ω and α C 1 Ω, the following identity holds: λ α n, div u Ω + µ α n, εu Ω = λdiv α, div u Ω + µdiv α, εu Ω + λ Re div u, div vα + 4µ Re εu, ε vα Ω Ω d d λ Re div u, α j u i x i=1 j=1 i x j Ω d d d ui µ Re + u j, α k u i x j x i x j x k i=1 j=1 k=1 Lemma.3.. For u H Ω and α C 1 Ω, the following identity holds:. Ω div α, u Ω = α n, u Ω Reu, uα Ω. Similar to other analysis involving the stress tensor σ, it is necessary to use the well-known Korn s inequality to obtain the desired generalized weak coercivity property. It is stated here as a lemma. For a proof, see [63]. Lemma.3.3. There exists a positive constant K such that for any v H 1 Ω the following inequality holds: [ ] v H 1 Ω K εu L Ω + v L Ω. As was the case in Section., the analysis used in this section will follow closely to that in [7]. In [7] the authors found it necessary to use a Korn-type inequality on the boundary Ω to obtain estimates that are optimal in terms of the frequency ω. This Korn-type inequality still remains a conjecture. As stated in [7], this conjecture 1

38 is believed to hold for the solution of the elastic Helmholtz problem since a similar result is shown in [8] for the solution of the Lamé systems of elastostatics. Conjecture.3.4. There exists a positive constant K such that for any u H Ω the following Korn-type inequality holds: u L Ω K [ ] u L Ω + εu L Ω..4 With these technical lemmas in hand, we have all the tools necessary to prove a generalized weak coercivity property on a,. This will be done in two steps. First, we prove a preliminary result that does not make use of Conjecture.3.4. Next, we prove a generalized weak coercivity property for u in a more restrictive function space i.e. the space on which Conjecture.3.4 holds. Lemma.3.5. Let Ω be a generalized star-shape domain such that there exists α C 1 Ω satisfying Then for all u H Ω and ɛ > 0 there holds u E ɛ u L Ω c + µ u L Ω + c +µ εu L Ω + Re au, uα + 1 dc 1 u + 1 Rωρ + R ωc A + Im au, u, c A ɛ where E is defined by u E := c 1 ω ρ u L Ω + c 1λ div u L Ω + c 1µ εu L Ω + c + µ u L Ω + c + λ div u L Ω + c +µ εu L Ω. Proof. In a manner similar to the proof of Theorem..4, setting v = u in.3 and taking both real and imaginary parts separately we get Re au, u = λ div u L Ω + µ εu L Ω ω ρ u L Ω,.5 Im au, u = ω Au, u Ω..6

39 Again, it is clear that this first test function alone cannot yield the desired result because of the sign difference in.5. For this reason, a second test function, motivated by our Rellich identities, will be selected. Let v = uα for the rest of this proof. Substituting this test function into.3 and multiplying through by gives Re au, v = λdiv u, div v Ω + 4µεu, εv Ω Re ω ρu, v Ω ω Im Au, v Ω. By the Rellich identities for the elastic Helmholtz operator i.e. Lemmas.3.1 and.3., we get Re au, v = λ α n, div u + µ α n, εu λ div α, div u Ω Ω Ω µ div α, εu d d + λ Re div u, α j u i Ω x i=1 j=1 i x j Ω d d d ui + µ Re + u j, α k u i x j x i x j x k i=1 j=1 k=1 ω ρ α n, u Ω + ω ρ div α, u Ω ω Im Au, v Ω. Ω Equivalently, ω ρ div α, u λ div α, div u µ div α, εu Ω Ω Ω d d + λ Re div u, α j u i x i=1 j=1 i x j Ω d d d ui + µ Re + u j, α k u i x j x i x j x k i=1 j=1 k=1 = ω ρ α n, u Ω λ α n, div u Ω µ α n, εu Ω + ω Im Au, v Ω + Re au, v. Ω 3

40 Applying the properties of α from to the above identity produces dc 1 ω ρ u L Ω dc 1λ div u L Ω dc 1µ εu L Ω d + c 1 λ Re div u, u d d i ui + c 1 µ Re + u j, u i x i Ω x j x i x j Ω i=1 i=1 j=1 Rω ρ u L Ω c +λ div u L Ω c +µ εu L Ω + ω Im Au, v Ω + Re au, v..7 At this stage, we appeal to the following simplifications: c 1 λ Re d i=1 div u, u i = c 1 λ Re div u, x i Ω = c 1 λ div u L Ω, d i=1 u i x i Ω and d d ui c 1 µ Re + u j, u i x i=1 j=1 j x i x j Ω [ d d ui = c 1 µ Re + u j, u d i + x i=1 j=1 j x i x j Ω i=1 d d ui = c 1 µ Re + u j, u i + u j x j x i x j x i i=1 j=1 = 4c 1 µ εu L Ω. Ω d j=1 ui + u j, u ] j x j x i x i Ω We apply these simplifications to.7 to get dc 1 ω ρ u L Ω + dc 1λ div u L Ω + dc 1µ εu L Ω Rω ρ u L Ω c +λ div u L Ω c +µ εu L Ω + ω Im Au, v Ω + Re au, v. 4

41 Note, for d >, the terms with coefficient d are negative. To eliminate these terms from the left hand side LHS, add d c 1 times.5 to the above identity to obtain the following: c 1 ω ρ u L Ω Rω ρ u L Ω c +λ div u L Ω c +µ εu L Ω + ω Im Au, v Ω + Re au, v + dc 1 u. We notice that.6 allows us control over the terms on the right hand side RHS involving u L Ω. With this in mind, we apply the Cauchy-Schwarz inequality along with Young s inequality to ω Im Au, v Ω to obtain c 1 ω ρ u L Ω Rω ρ u L Ω c +λ div u L Ω c +µ εu L Ω + ω Au L Ω v L Ω + Re au, v + dc 1 u Rω ρ u L Ω c +λ div u L Ω c +µ εu L Ω + Rω Au L Ω u L Ω + Re au, v + dc 1 u Rω ρ u L Ω c +λ div u L Ω c +µ εu L Ω The term ɛ u L Ω + R ω C A u L ɛ Ω + ɛ u L Ω + Re au, v + dc 1u. will be controlled later using the boundary Korn-type inequality c.f. Conjecture.3.4. With this in mind, we add and subtract 5

42 c + µ u L Ω and apply.6 to yield c 1 ω ρ u L Ω + c +µ u L Ω c +λ div u L Ω c +µ εu L Ω + ɛ u L Ω c + µ u Ω + c + µ εu Ω + Re au, v + dc 1 u + c + λ div u L Ω c +µ εu L Ω Rω ρ + R ω C A + c + µ ɛ + ɛ u L Ω c + µ u Ω + c + µ εu Ω + Re au, v + dc 1 u + 1 c A Rωρ + R ωc A + c +µ ɛ ω u L Ω Im au, u. To obtain a norm on H 1 Ω on the LHS, we subtract.5 from the above inequality, and move some terms to the LHS to get c 1 ω ρ u L Ω + c 1λ div u L Ω + c 1µ εu L Ω + c + µ u L Ω + c +λ div u L Ω + c +µ εu L Ω ɛ u L Ω c + µ u Ω + c + µ εu Ω + Re au, v + 1 dc 1 u + 1 Rωρ + R ωc A + c +µ Im au, u. c A ɛ ω Therefore, the assertion holds. In order to prove a generalized weak coercivity property on a,, an estimate to control the term ɛ u L Ω on the RHS of the inequality in Lemma.3.5 needs to be established. This is where a Korn-type inequality on the boundary would be helpful. With this in mind, we introduce the special function spaces { } V := v H 1 Ω εv L Ω, { V K := u V u L Ω K [ u L Ω + εu L Ω ] }, 6

43 where K is a positive constant. With the help of these spaces, the following generalized weak coercivity property holds. Theorem.3.6. Let Ω be a domain for which there exists α C 1 Ω satisfying Then for any K > 0 and u V K the following inequality holds: where Im au, v Re au, v sup + sup v V v E v H 1 Ω v L Ω { [ γ := max 4R K M := 1 c A 1 + ω ρ µ Rωρ + R ωc A K c + µ 1 γ u E, + 4R K + 1 d c 1 + c +µ ω, ] 1 }, M, u E := c 1 ω ρ u L Ω + c 1λ div u L Ω + c 1µ εu L Ω + c + µ u L Ω + c + λ div u L Ω + c +µ εu L Ω u L Ω := c 1ω ρ u L Ω + c +µ u L Ω. Proof. As was the case in the proof of Theorem..4, we only give a proof for u V K H Ω. After we prove the result for this more restrictive case, we can use a limiting process to yield the result for u V K. By Lemma.3.5 with ɛ = c +µ K we obtain u E c +µ K u L Ω c + µ u L Ω + c +µ εu L Ω + Re au, ˆv + 1 Rωρ + R ωc A K + c +µ Im au, u, c A c + µ ω 7

44 where ˆv := uα + 1 dc 1 u. With this choice of ɛ, there holds c + µ K u L Ω c + µ u L Ω + c +µ εu L Ω c + µ u L Ω + εu L Ω c+ µ u Ω + c + µ εu L Ω 0. Thus, u E Re au, ˆv + M Im au, u..8 Next, by the definitions of E and L Ω we get ˆv L Ω = c 1ω ρ ˆv L Ω + c +µ ˆv L Ω.9 4c 1 R ω ρ u L Ω + 4R c + µ u L Ω + 1 d c 3 1ω ρ u L Ω + 1 d c 1c + µ u L Ω [ ] 4R K 1 + ω ρ + 4R K + 1 d c 1 u E µ γ u E. It follows from.8 and.9 that Im au, v Re au, v sup + sup v V v E vh 1 Ω v L Ω Im au, u u E + Re au, ˆv ˆv L Ω Re au, ˆv + M Im au, u M u E γ u E M Im au, u + Re au, ˆv γ u E 1 γ u E. Thus, the desired generalized weak coercivity property holds. 8

45 .4 The Time-Harmonic Maxwell Operator As was the case in Sections. and.3, we begin this section by defining a generalized star-shape domain that is specially suited to the time-harmonic Maxwell operator. In this section, the restriction that d = 3 i.e. Ω R 3 is assumed. Similar to section.3, the domain Ω is also restricted to the case where a scattering portion of the boundary is not present. That is, Ω = and thus Ω + = Ω. Lastly, in this section, Ω is defined to be a generalized star-shape domain such that there exists α C 1 Ω satisfying the following properties: α i = α i x i in Ω and for i = 1,, 3,.30 α R in Ω,.31 α n c + > 0 on Ω,.3 { } αi div α max c 1 > 0 in Ω..33 i=1,,3 x i Maxwell s equations are defined using the curl operator. For this reason, some special function spaces need to be defined on Ω before a weak formulation can be defined. { } Hcurl, Ω := v L Ω curl v L Ω, { } Hdiv, Ω := v L Ω div v L Ω, { } Hdiv 0, Ω := v L Ω div v = 0, { } V := v Hcurl, Ω v L Ω, { } ˆV := v Hcurl, Ω curl v Hcurl, Ω and v Hcurl, Ω. Following the example set forth in Sections. and.3, the weak formulation of is derived below. Multiplying 1.6 with a smooth test function v C Ω 9

46 and integrating over the domain Ω gives curl curl E, v Ω k E, v Ω = f, v Ω..34 In order to derive the appropriate integration by parts formula, we start with the following identity: div curl E v = curl curl E v curl E curl v..35 This identity is easily obtained from the well-known vector calculus identity: div a b = b curl a a curl b along with the divergence theorem yields the following integration by parts identity: curl curl E, v Ω = curl E, curl v Ω + div curl E vdx.37 = curl E, curl v Ω + curl E v, n Ω Ω = curl E, curl v Ω curl E n, v Ω = curl E, curl v Ω curl E n, v T Ω. Here, the identities a b c = c a b and a b = b a along with the decomposition v = v T + v nn and the fact a n n = 0 have been used. By the boundary conditions 1.7 we get curl curl E, v Ω = curl E, curl v Ω iλ E T, v T g, v Ω T Ω. 30

47 div α, curl u Ω + α n, curl u Ω Applying the above identity to.34 yields the following weak formulation of the time-harmonic Maxwell s equations: find E V such that ae, v = f, v Ω + g, v T Ω v V,.38 where the sesquilinear form a, on V V is defined by au, v := curl u, curl v Ω k u, v Ω iλ u T, v T Ω..39 After having derived the above weak formulation for the time-harmonic Maxwell s equations, the focus of this section shifts to the goal of obtaining a generalized weak coercivity property for the sesquilinear form a,. Again, Rellich identities will be used to achieve this goal. These Rellich identities are generalizations of those that can be found in [39]. Similar identities are derived and used to achieve stability estimates for the time-harmonic Maxwell s equations when Ω is a star-shape domain c.f. [50]. Since the general case of α being a C 1 Ω function was not considered, detailed proofs for these Rellich identities are given below. The following notation will be used in the Rellich identities: a b := a b. Lemma.4.1. Suppose u H Ω and α C 1 Ω. Then the following identity holds: = Re curl u, curl curl u α Ω + Re curl u, curl u α Ω. Proof. To prove this lemma two additional differential identities along with the divergence theorem are used. Set v = curl u α. To derive the first identity, 31

48 we recall the following well-known identity involving the curl operator: curl v = curl curl u α = div αcurl u αdiv curl u + α curl u curl u α. Using the fact that div curl u = 0, the above identity gives the first sought-after identity: curl v = div αcurl u + α curl u curl u α..40 To derive the second sought-after identity, expanding div a b using the product rule for the divergence and gradient yields div a b = div a b + a b b.41 = div a b + b a b + b a b = div a b + Re b a b..41 immediately gives the second sought-after identity div α curl u = div α curl u + Re curl u α curl u..4 Taking the complex conjugate of.40, applying the dot product with curl u, and taking the real part gives Re curl u curl v = div α curl u + Re curl u α curl u Re curl u curl u α, 3

49 which together with.4 gives Re curl u curl v = div α curl u div α curl u + div α curl u Re curl u curl u α = div α curl u + div α curl u Re curl u curl u α. Integrating the above identity over Ω and using the divergence theorem yields Recurl u, curl v Ω = div α, curl u + div α curl u dx Ω Re curl u, curl u α Ω = div α, curl u Ω + α n, curl u Ω Re curl u, curl u α Ω. By rearranging terms and recalling v = curl u α, the desired Rellich identity is obtained. Lemma.4.. Suppose u H 1 Ω and α C 1 Ω. Then the following identity holds: Reu, curl u α Ω + 1 div α, u + 1 Ω α n, u Ω = Reαdiv u, u Ω + Re u, u α Ω Re α u, u n Ω. Proof. Like the proof of Lemma.4.1, this proof relies on two differential identities along with integration by parts. The first differential identity is the following identity involving the curl operator: curl α u = αdiv u udiv α + u α α u

50 The next identity follows from.41 and reads as div α u = div α u + Re u α u..44 By.36 and the divergence theorem we get curl u, v Ω = u, curl v Ω + div u vdx.45 = u, curl v Ω + u v, n Ω. Ω that It follows from the identities a b c = b c a,.45,.43, and.44 Reu, curl u α Ω = Recurl u, α u = Re u α u, n Ω + Reu, curl α u Ω = Re α u, u n Ω + Reu, αdiv u Ω div α, u Ω + Re u, u α Ω Re u, α u Ω = Re α u, u n Ω + Reu, αdiv u Ω div α, u Ω + Re u, u α Ω 1 α n, u Ω + 1 div α, u Ω = Re α u, u n Ω + Reu, αdiv u Ω 1 div α, u Ω + Re u, u α Ω 1 α n, u. Ω Thus, the desired Rellich identity is obtained by rearranging the terms above. The above Rellich identities can be used to establish a generalized weak coercivity property for a,. Note that a generalized weak coercivity property for a, was already established in [43] for a star-shape domain and what is below is an extension of that result to a generalized star-shape domain. 34

51 Theorem.4.3. Let Ω be a generalized star-shape domain such that there exists α C 1 Ω satisfying Then for any u V Hdiv 0, Ω the following generalized weak coercivity property holds on a, : Im au, v sup v ˆV v E Re au, v + sup v V v L Ω 1 γ u E, where γ := 4kR + M, M := Rc + + R k + R λ, λc + 1 u L Ω := c 1 k u L Ω + c +k u L Ω, u E := c 1 k u L Ω + c 1 curl u L Ω + c +k u L Ω + c + curl u L Ω 1. Proof. Similar to the proof of Theorems..4 and.3.6, this proof makes use of two specific test functions. The first is v = u. Using this test function in.39 and taking the real and imaginary parts separately yield Re au, u = curl u L Ω k u L Ω,.46 Im au, u = λ u T L Ω..47 The second test function is v = curl u α motivated by the Rellich identities. Recall that α is the vector field defined by the generalized star-shape condition on Ω. Using Lemmas.4.1 and.4. gives Re au, v = Recurl u, curl v Ω k Reu, v Ω + λ Im u T, v T Ω.48 = div α, curl u Ω Recurl u, curl u α Ω + α n, curl u Ω + k div α, u Ω k Reu, u α Ω + k α n, u Ω + k Re α u, u n Ω + λ Re u T, v T Ω. Here the fact that div u = 0 has been used. 35

52 By using.30 we get u u α = 3 u i u α i = i=1 u max i=1,,3 { αi x i }. 3 i=1 3 u i j=1 u j α i = x j 3 i=1 3 j=1 u i α i x i.49 Similarly, curl u curl u α curl u max i=1,,3 Combining.49 and.50 with.33 gives { αi x i k div α, u Ω k Reu, u α Ω k div α max i=1,,3 c 1 k u L Ω }..50 { αi x i }, u Ω and div α, curl u Ω Recurl u, curl u α Ω c 1 curl u L Ω. Rearranging the terms in.48 and substituting the above inequalities yield c 1 k u L Ω + c 1 curl u L Ω k α n, u Ω α n, curl u Ω + k α u, u n Ω λ u T, v T Ω + Re au, v

53 To bound the term k α u, u n Ω, by the decomposition α = α T + α nα and the identity a b c d = a cb d a db c we get k α u, u n Ω = k α T u, u n Ω k α nn u, u n Ω = k α T u, u n Ω + k α T n, u Ω + k α n, u n Ω = k α T u T, u n Ω + k α n, u n Ω. This identity allows us to rewrite.51 as c 1 k u L Ω + c 1 curl u L Ω.5 k α n, u Ω α n, curl u Ω + k Re α n, u n Ω k Re u n, α T u T Ω λ Re u T, v T Ω + Re au, v. Noting that u T = n u n = u n u n n = u n on Ω. 37

54 We use this identity along with.31,.3, Cauchy Schwarz and Young s inequality in.5 to get c 1 k u L Ω + c 1 curl u L Ω c + k u L Ω c + curl u L Ω + Rk u T L Ω + Rk u L Ω u T L Ω + Rλ u T L Ω curl u L Ω + Re au, v c + k u L Ω c + curl u L Ω + Rk u T L Ω + c +k u L Ω + R k u T L Ω c + + R λ c + u T L Ω + c + curl u L Ω + Re au, v c + k u L Ω c + curl u L Ω + Re au, v + Rc + + R k + R λ c + u T L Ω. Multiplying the above inequality by and making use of.47 yield c 1 k u L Ω + c 1 curl u L Ω + c +k u L Ω + c + curl u L Ω + 4c +k u T L Ω Rc + + R k + R λ λ u T L λc Ω + Re au, 4v. + M Im au, u + Re au, 4v. Thus, u E M Im au, u + Re au, 4v

55 By the definitions of v, E, and L Ω we have v L Ω = c 1 k curl u α L Ω + c +k curl u α L Ω 1 c 1 R k curl u L Ω + c +R k curl u L Ω kr u E It follows from.53 and.54 that Im au, v Re au, v sup + sup v H Ω v E v H 1 Ω v L Ω 1 γ Im au, u u E + M Im au, u M u E Re au, v 4curl u α L Ω + Re au, v γ u E M Im au, u + Re au, v u E 1 γ u E, which yields the desired generalized weak coercivity property..5 Applications to Stability Estimates In Sections.,.3, and.4, it was demonstrated that each Helmholtz-type problem satisfies a generalized weak coercivity property. The goal of this section is to give one application of the generalized weak coercivity properties. Namely, we apply the generalized weak coercivity properties to derive wave-number explicit solution estimates for each Helmholtz-type problem. These solution estimates are stated in the following three theorems. Theorem.5.1. Let Ω R d be a generalized star-shape domain with α C 1 Ω satisfying..8. Suppose u V solves.9 with f L Ω and g L Ω +. 39

56 Then the following estimate holds: u E γβ f L Ω + g L Ω +, where { [c γ := max 4c 3 + c k + 1R ] } 1, M M := kr + kr + c + c + k β := 1 c4 k + 1 c+,, 1 u L Ω := k c 4 u L Ω + c + u L Ω +, u E := k c 4 u L Ω + c 4 u L Ω + c + u L Ω + + c + u L Ω + + c u L Ω 1., Proof. Let v H 1 Ω. The Cauchy-Schwarz inequality yields the following series of inequalities: au, v = f, v Ω + g, v Ω+ f L Ω v L Ω + g L Ω + v L Ω + 1 f L Ω + g L Ω + v L Ω + v L Ω + 1 c4 k + 1 f L Ω + g L Ω + c+ 1 1 v L Ω. Similarly, for v V we have au, v = f, v Ω + g, v Ω+ 1 c4 k f L Ω + g L Ω v E. c+ 40

57 Thus, 1 γ u Im au, v Re au, v E sup + sup v H Ω v E v H 1 Ω v L Ω sup v H Ω 1 c4 k + 1 au, v v E + sup v H 1 Ω au, v v L Ω c+ f L Ω + g L Ω c4 k f L Ω + g L Ω + c+ β f L Ω + g L Ω +. The proof is complete. Theorem.5.. Let Ω R d be a generalized star-shape domain such that there exists α C 1 Ω satisfying Suppose that there exists some positive constant K such that u V K solves. for f L Ω and g L Ω. Then the following stability estimate holds: where u E γ { [ γ := max 4R K M := 1 c A 1 ω + 1 f L c 1 c+ µ Ω + g L Ω, 1 + ω ρ µ Rωρ + R ωc A K c + µ + 4R K + 1 d c 1 + c +µ ω, ] 1 }, M, u E := c 1 ω ρ u L Ω + c 1λ div u L Ω + c 1µ εu L Ω + c + µ u L Ω + c +λ div u L Ω + c +µ εu L Ω, u L Ω := c 1ω ρ u L Ω + c +µ u L Ω. Proof. This proof is very similar to that of Theorem.5.1. We begin with finding an upper bound on au, v for some v H 1 Ω. By the Cauchy-Schwarz inequality, 41

58 we get au, v = f, v Ω + g, v Ω f L Ω v L Ω + g L Ω v L Ω 1 f L Ω + g L Ω v L Ω + v L Ω 1 c 1 ω + 1 c + µ 1 1 f L Ω + g L Ω c 1 ω v L Ω + c +µ v L Ω. 1 Therefore, for v H 1 Ω, au, v 1 c 1 ω f L c + µ Ω + g L Ω v L Ω..55 By noting that, v L Ω v E, we find au, v 1 c 1 ω f L c + µ Ω + g L Ω v E..56 These bounds on a, in conjunction with Theorem.3.6 imply that 1 γ u E sup v H Ω Im au, v Re au, v + sup v E v H 1 Ω v E au, v sup v H Ω Hence the stability estimate holds. + sup v E v H 1 Ω 1 c 1 ω + 1 c + µ au, v v E 1 f L Ω + g L Ω Theorem.5.3. Let Ω R 3 be a generalized star-shape domain with α C 1 Ω satisfying Suppose E ˆV solves.38 for f Hdiv, Ω, g L Ω.. 4

59 Then the following solution estimate on E holds: E E 4γ k f LΩ + g L Ω c 1 c + + c 1 + R 1 k f Ω + R 1 k div f L Ω where γ := 4kR + M, M := Rc + + R k + R λ, λc + 1 u L Ω := c 1 k u L Ω + c +k u L Ω, u E := c 1 k u L Ω + c 1 curl u L Ω + c +k u L Ω + c + curl u L Ω 1. Proof. This proof follows the proof of Theorem.3 from [43] with changes in some details dealing with the generalized star-shape condition imposed on the domain Ω. Now since E ˆV solves.38 then E satisfies 1.6 a.e. in Ω and we find k div E = div curl curl E k E = div f a.e. in Ω. This implies div E = k div f a.e. in Ω. To apply Theorem.4.3 we need a vector field u H Ω Hdiv 0, Ω. Therefore, unlike the proofs of Theorems.5.1 and.5., we cannot directly apply Theorem.4.3 to E. To overcome this difficulty, consider an auxiliary vector field F = φ, where φ H0Ω 1 solves the following Poisson equation: φ = k div f a.e. in Ω..57 By definition, div F = k div f a.e. in Ω. Also, the definition of F ensures curl F = 0 so F ˆV. Thus for u := E + F, u ˆV Hdiv 0, Ω. With this in mind, the estimate on E will be obtained by estimating F and u separately. Estimates on F 43

60 can be obtained based on its definition and estimates on u can be obtained from the generalized weak coercivity property for the time-harmonic Maxwell operator. To derive estimates for F, we test.57 with φ and integrate by parts to obtain φ L Ω = k f, φ Ω k f L Ω φ L Ω. Hence, F L Ω = φ L Ω k f L Ω..58 Next, testing.57 by the test function φ α = F α and applying integration by parts and Lemma..3 we obtain Re φ, φ α Ω = Re φ, φ α Ω = Re φ, φ α φ + Re Ω n, φ α = div α, φ Ω Re 3 i=1 3 j=1 φ, α j φ x i x i x j α n, φ Ω + Re φ n, φ α Ω. Ω Ω Now F T = φ T = 0 on Ω since φ H 1 0Ω. Using this fact along with.30,.3, and.33 in the above inequality gives Re φ, φ α Ω = div α, φ 3 Re α j, φ Ω x i=1 i x j Ω α n, φ φ + Re Ω n, φ T + φ n n α Ω { } αi div α max, φ + α n, φ i=1,,3 x Ω i c 1 F L Ω + c + F Ω. Ω 44

61 that Therefore, it follows from.57,.58,.31, and the Cauchy-Schwarz inequality c + F L Ω = c + φ L Ω k Rediv f, F α Ω Rk div f L Ω F L Ω Rk 4 div f L Ω f L Ω Rk 4 div f L Ω + Rk 4 f L Ω..59 Since curl F = 0 in Ω,.58 and.59 yield the following estimate for F E : F E = c 1 k F L Ω + c + k F L Ω c 1 k f Ω + Rk div f L Ω + Rk f L Ω = c 1 + R k f Ω + R k div f L Ω..60 Next, we derive estimates for u. Note that since φ H 1 0Ω and F = φ, F T = 0 on Ω and curl F = 0 in Ω. These two facts imply that u satisfies the following weak form of the time-harmonic Maxwell s equations: au, v = f k F, v Ω + g, v T L Ω v V

62 where a, is the sesquilinear form defined in.39. Thus, by the Cauchy-Schwarz inequality,.58, and.61 we get au, v = f k F, v Ω + g, v T Ω f, v Ω + k F, v Ω + g, v T Ω f L Ω v L Ω + k F L Ω v L Ω + g L Ω v T L Ω = f L Ω v L Ω + g L Ω v T L Ω f L Ω + g L Ω v L Ω + v T L Ω 1 k f LΩ + g L Ω c 1 k v L c Ω + c +k v L Ω 1 c + 1 = k f LΩ + g L Ω v L c 1 c Ω. + Thus for v H 1 Ω, 1 au, v k f LΩ + g L Ω v L c 1 c Ω Note for v ˆV, v L Ω v E. Hence, 1 au, v k f LΩ + g L Ω v E..63 c 1 c + Substituting.6 and.63 into the generalized weak coercivity condition given in Theorem.4.3 gives 1 γ u E sup v H Ω Im au, v Re au, v + sup v E v H 1 Ω v E au, v sup v H Ω 4k 1 1 c c + au, v + sup v E v H 1 Ω v E f LΩ + g L Ω. 46

63 Thus, u E 4γ k f LΩ + g L Ω..64 c 1 c + Recall E = u F. Thus.60 and.64 yield E E 4γ k f LΩ + g L Ω c 1 c + + c 1 + R 1 k f Ω + R 1 k div f L Ω. Remark.5.4. a The above solution estimates ensure uniqueness of the solution to each Helmholtz-type problem in their respective solution spaces. bthe adjoint problem for each Helmholtz-type problem differs only in the sign of the boundary integral terms. For this reason, all of the results of this chapter also hold for these adjoint problems. In particular, the uniqueness results. By the Fredholm Alternative Principle this ensures existence of the solutions to the Helmholtz-type problems in their respective solution spaces. 47

64 Chapter 3 Absolutely Stable Discontinuous Galerkin Methods for the Elastic Helmholtz Equations As is the case for the scalar Helmholtz equation, the angular frequency ω plays a key role in the analysis and implementation of any numerical method used to solve the elastic Helmholtz equations. It is a well-known fact that in order to resolve the wave numerically one must use some minimum number of grid points in each wave length l = π/ω in every coordinate direction. This yields the minimum mesh constraint ωh = O1, where h is the mesh size parameter. In fact, the widely held rule of thumb is to use 6 1 grid points per wave length. In [54], Babǔska et al proved the necessity of this rule of thumb in the 1-dimensional case for the scalar Helmholtz equation. In [54], it was also shown that the H 1 error bound for the finite element solution contains a pollution term that contributes to the loss of stability for this method in the case of a large ω. The pollution term also causes the error to increase as ω increases under the mesh constraint ωh = O1. This forces one to adopt a more stringent mesh condition to guarantee an accurate numerical solution for high frequency waves. 48

65 The loss of stability of the standard finite element method applied to Helmholtztype problems is an important issue to address and a fundamental limitation to overcome. Specifically, in [6, 9, 30], a strict mesh condition of ω h = O1 called the asymptotic mesh constraint was required to obtain optimal and quasi-optimal error estimates for finite element approximations applied to the scalar Helmholtz equation. In [6], this same mesh condition was used to obtain error estimates for the elastic Helmholtz equations. Requiring such a stringent mesh constraint makes the use of a practical coarse mesh space impossible in the case that ω is large. This is a hurdle that must be overcome if one wishes to use multi-level algebraic solvers, such as the multi-grid method or multi-level domain decomposition method. Thus, it is the goal of this chapter to develop and analyze an interior penalty discontinuous Galerkin IP-DG method that will be absolutely stable for the elastic Helmholtz equations. In other words, a method in which a-priori solution estimates can be obtained for any ω, h > 0. This chapter follows the example of [4, 79, 43] which give similar methods for the scalar Helmholtz equation and the time-harmonic Maxwell s equations. Section 3.1 introduces standard notation required to formulate a discontinuous Galerkin method and presents the IP-DG method. Also, in this section, some key properties of the proposed method are demonstrated. In Section 3., error estimates are obtained for the asymptotic mesh regime i.e. when ω h C. To accomplish this, we define and analyze a specific elliptic projection operator for the elastic Helmholtz equations. With this projection operator, Schatz argument is carried out to obtain the optimal error estimates. Section 3.3 is devoted to establishing stability and error estimates for the pre-asymptotic mesh regime i.e. when ω h > C. This is an important feature of the IP-DG method proposed in this chapter since it has not been shown that previous discretization techniques yield stability in this mesh regime. Section 3.4 is devoted to numerical experiments that validate properties of the proposed IP-DG method and compare it to the standard finite element method. 49

66 3.1 Formulation of the IP-DG Method In this section, an interior penalty discontinuous Galerkin IP-DG method for the elastic Helmholtz equations is formulated. This formulation will follow those in [4, 44]. The methods referenced in the previous papers are absolutely stable, i.e. stable for all ω, h > 0 a trait sought in the discretization methods for Helmholtz-type problems. First, some standard IP-DG notation needs to be introduced. Let T h be a shape regular partition of the domain Ω, such that for each cell K T h, h K := diamk. Also, for each edge/face e of a cell K, define h e := diame. T h is called shape regular if there exist positive constants m 1, m such that for any K T h and e an edge/face of K the following inequality holds: m 1 h e h K m h e. The partition T h is parameterized by h, which denotes the maximum spatial cell size, i.e. h := max K Th {h K }. We note that the discontinuous Galerkin DG methodology allows greater flexibility in terms of meshing the domain Ω. In particular, one can use any polyhedral elements in the partition. In some cases, the partition is made of elements with curved boundaries. Let E I h := set of all interior edges/faces of T h, E B h := set of all boundary edges/faces of T h on Ω. 50

67 For any edge e Eh I, let K e, K e T h such that e = K e K e. For such an edge e, we define the following jump and average operators: v Ke v K e, if the global labeling number of K e is greater than that of K e, [v] e := v K e v Ke, if the global labeling number of K e is greater than that of K e, and {v} e := 1 v Ke + v K e. For e Eh B, we use the convention [v] e = {v} e := v e. Also keeping in mind the idea of cell by cell integration by parts, the outward normal vector n e to e Eh I will need to be defined. Let n e be the unit outward normal vector to K e on e, where e = K e K e and K e has a bigger global labeling number than that of K e. Define the DG energy space E as E := H K. K T h Note that unlike a conforming finite element method, this energy space can be discontinuous across cell boundaries. Multiplying 1.4 by some v E and integrating 51

68 by parts piecewisely we get f, v Ω = div σu, v Ω ω ρu, v Ω = K T h div σu, v K ω ρu, v Ω = K T h λdiv u, div vk + µεu, εv K ω ρu, v Ω K T h σun K, v K = K T h λdiv u, div vk + µεu, εv K ω ρu, v Ω e E I h e[σun e v]ds e E B h σun e, v e. To the above identity we apply 1.5 along with a well-known identity concerning the jump of a product, i.e. [a b] = {a} [b] + [a] {b}, and get f, v Ω + g, v Ω = K T h λdiv u, div vk + µεu, εv K ω ρu, v Ω e E I h {σune }, [v] e + [σun e ], {v} e + iω Au, v e. Now assuming that the solution u of is smooth enough, we find [σun e ] = [u] = 0 on all e Eh I. Thus the above identity is equivalent to f, v Ω + g, v Ω = K T h λdiv u, div vk + µεu, εv K ω ρu, v Ω 3.1 e E I h 3. {σune }, [v] e + η [u], {σvn e } e + iω Au, v Ω. The term η [u], {σvn e } e is introduced as a possible avenue toward symmetrizing the RHS. Thus, η is called a symmetrization parameter. It is standard for one 5

69 to consider three possible values for η. These values are η = 1 for a symmetric formulation, η = 0, and η = 1 for an anti-symmetric formulation. In this chapter, we only focus on the symmetric case with η = 1. Here it is left as a variable parameter to offer different formulations of this method. An important aspect of the IP-DG methods is the use of penalty terms to ensure the coercivity of the sesquilinear forms involved in the formulation. To this end, we introduce two penalty sesquilinear forms J 0, and J 1,. w, v E as J 0 w, v := e E I h J 1 w, v := e E I h γ 0,e h e [w], [v] e, γ 1,e h e [σwne ], [σvn e ] e, They are defined for where γ 0,e, γ 1,e > 0 are called the penalty parameters for e Eh I. Note that by the smoothness of the solution u we have J 0 u, v = J 1 u, v = 0. Thus, 3. can be rewritten as f, v Ω + g, v Ω = K T h λdiv u, div vk + µεu, εv K ω ρu, v Ω e E I h {σune }, [v] e + η [u], {σvn e } e + i J 0 u, v + J 1 u, v + iω Au, v Ω. Therefore, u satisfies A h u, v = f, v Ω + g, v Ω v E,

70 where A h, is defined on E E as A h w, v := a h w, v ω ρw, v Ω + iω Aw, v Ω, 3.4 a h w, v := λ div w, div v + µ εw, εv K K K T h {σwne }, [v] e + η [w], {σvn e } e e E I h + i J 0 w, v + J 1 w, v. 3.5 With the DG sesquilinear form A h, in hand, a discrete function space is needed to formulate the IP-DG method. For this chapter, only piecewise linear polynomial functions over the partition T h will be considered. Thus the IP-DG approximation space V h is defined as V h := P 1 K. K T h With all the building blocks in place, our IP-DG method is defined by seeking u h V h such that A h u h, v h = f, v h + g, v Ω h v Ω h V h Some Properties of the IP-DG Method In this subsection, some useful properties of the above IP-DG method 3.6 are established. From this point on, we only consider the case η = 1 for simplicity. Also, we assume there exists a positive constant C such that h K h Ch K K T h, γ 0,e γ 0 Cγ 0,e e E I h, γ 1,e γ 1 Cγ 1,e e E I h. 54

71 The above constraints are not necessary for the analysis of this chapter, but rather are in place to make the constants in the inequalities more tractable. In order to analyze the proposed IP-DG method, we introduce the following seminorms: 1 v 1,h := λ div v L K + µ εv L K, K T h 1 v 1,h := v 1,h + J 0 v, v + J 1 v, v, v 1,h := v 1,h + 1 h e {σvn e } L γ e. 0,e e Eh I Note on V h, the semi-norms 1,h and 1,h are equivalent. This is trivial since V h is a finite dimensional vector space. On the other hand, it can be shown that this equivalence is independent of the dimension of V h. This result is non-trivial, and thus it is proved in a lemma below. To prove the equivalence, we need two inequalities which hold for polynomial functions. Namely, the inverse and trace inequalities as given below: v h L e Ch 1 e v h L K v h V h, 3.7 σv h n e L e Ch 1 e σv h L K v h V h, 3.8 where K T h and e is an edge/face of K. These inequalities will be used throughout the rest of this chapter. Lemma For any v h V h, there holds v h 1,h v h 1,h Cξ 1 vh 1,h, where ξ := 1 + 1γ0 and C is a positive constant independent of ω, h, γ 0, γ 1. 55

72 Proof. Note that the first inequality is trivial. To show the second inequality we use 3.8 as follows: v h 1,h = v 1,h + e E I h v h 1,h + C e E I h h e γ 0,e {σvn e } L e v h Cλ + µ 1,h + C γ 0 h e γ 0,e 1 h e σv L K e + σv L K e K T h 1 + λ + µ v h γ 1,h. 0 λ div v h K + µ εv h L K In order to show that 3.6 is well posed, we need to verify that A h, is both continuous and weakly coercive on V h. Weak coercivity in this case relies on the fact that V h is made up of piecewise linear polynomials. This fact infers the following lemma. Lemma For any 0 < δ < 1 and v h V h, v h 1,h δω ρ v h L Ω + C δλ + µ ω v h L ωh Ω + C δλ + µ J 0 v h, v h + C δ J γ 0 ω ρh 1 v h, v h. 3.9 γ 1 Proof. Note that v h K P 1 K for all K T h and thus div σv h K = 0. For any w h, v h V h and K T h, integrating by parts yields 0 = div σv h, w h K = σv h n K, w h K + λ div v h, div w h K + µ εv h, εw h K. 56

73 Thus, λ div v h, div w h K + µ εv h, εw h K = σv h n K, w h K. Setting w h = v h and summing over all K T h gives v h 1,h = K T h σvh n K, v h K = e E I h e E I h {σvh n e }, [v h ] e + [σv h n e ], {v h } e σvh n e, v h + e Eh B {σvh n e } L e [v h ] L e + [σv h n e ] L e {v h } L e + e E B h σv h n e L e v h L e. By the trace and inverse inequalities 3.7 and 3.8 along with the Cauchy-Schwarz inequality we obtain e v h 1,h C e E B h + C e E I h + C e E I h h 1 e σv h L K e v h L e h 1 e [v h ] L e σvh L K e + σv h L K e h 1 e [σv h n e ] L e vh L K e + v h L K e. 57

74 Finally, it follows from the discrete Cauchy-Schwarz inequality along with Young s inequality that v h 1,h Ch 1 + Cγ 1 0 σv h L K K T h e E I h + Cγ 1 1 h 1 1 v h L Ω γ 1 0,e 1 [v h ] L h e σv h L K e K T h e E I h γ 1,e h e [σv h n e ] L e 1 v h L Ω δ v h Cλ + µ 1,h + ω v h L δωh Ω + δ v h Cλ + µ 1,h + J 0 v h, v h δγ 0 + δ1 δω ρ v h L Ω + C J δ1 δω ρh 1 v h, v h. γ 1 Thus 3.9 holds. With this lemma in place, the following theorem establishes the continuity and weak coercivity of A h,. We note that since A is a constant symmetric positive definite SPD matrix that there exists positive constants c A, C A such that c A v L Ω Av, v Ω C A v L Ω v E Theorem The sesquilinear form A h, is continuous on the space E and weakly coercive on the space V h. That is, there exist positive constants M, C independent ω,h,γ 0,γ 1 such that A h w, v M 1 1 w 1,h + ω ρ w L Ω v 1,h + ω ρ v L Ω 3.11 for all w, v E and A h v h, v h C ξ + 1 ωh v ω h h 1,h + ω ρ v h L γ Ω, A h v h, v h J 0 v h, v h + J 1 v h, v h + ω Av h, v h Ω

75 for all v h V h. Here ξ = γ 0. Proof. To show 3.11, we appeal to the Cauchy-Schwarz and triangle inequalities. Thus, for any w, v E, we find A h w, v a h w, v + ω ρ w L Ω v L Ω w 1,h v 1,h + e E I h + e E I h {σwn e } L e [v] L e [w] L e {σvn e } L e + ω ρ w L Ω v L Ω w 1,h v 1,h + ω ρ w L Ω v L Ω + 1 he γ0,e {σwne } L γ e 0,e h e e Eh I γ0,e 1 he [w] L e 1 [v] L e 1 {σvne } L e + e E I h h e γ 0,e w 1,h v 1,h + +ω ρ w L Ω v L Ω 1 + e E I h h e + J 0 w, w {σwn e } L γ e 0,e e E I h h e J 0 v, v 1 {σvn e } L γ e 0,e h e w 1,h + J 0 w, w + {σwn e } L γ e + ω ρ w L Ω 0,e e Eh I M v 1,h + J 0 v, v + e E I h w 1,h + ω ρ w L Ω h e 1 {σvn e } L γ e + ω ρ v L Ω 0,e 1 v 1,h + ω ρ v L Ω

76 To verify weak coercivity, for any v h V h, taking the real and imaginary parts of A h v h, v h yields Re A h v h, v h = v h 1,h ω ρ v h L Ω + Re e E I h {σvh n e }, [v h ] e, 3.14 Im A h v h, v h = J 0 v h, v h + J 1 v h, v h + ω Av h, v h Ω Then 3.13 follows directly from To verify 3.1, we need to bound the term e Eh I {σvh n e }, [v h ]. This step e involves using the trace and inverse inequality and was already carried out in previous calculations c.f. Lemma Thus, Re A h v h, v h v h 1,h ω ρ v h L Ω + 1 Cγ 0 e E I h 3 v h 1,h ω ρ v h L Ω + C γ 0 J 0 v h, v h. γ 0 h e [v h ] L Ω 1 v h 1,h Combining the above inequality with 3.9 and using δ = 1 4 we get 1 v h 1,h + ω ρ v h L Ω Re A hv h, v h + v h 1,h + C γ 0 J 0 v h, v h Re A h v h, v h + 1 ω ρ v h L Ω + C ωc A v h L ωhc Ω A + C C J 0 v h, v h + J γ 0 ω ρh 1 v h, v h. γ 1 Thus, subtracting both sides of the above inequality by 1 ω ρ v h L Ω and using both 3.10 and 3.15 yield 1 v h 1,h + ω ρ v h L Ω Re A hv h, v h + C + 1 γ 0 Hence, 3.1 is verified. C γ 0 ωhc A + ωhc A 1 A ω ρh h v h, v h. γ 1 1 Im A ω ρh h v h, v h γ 1 60

77 Remark is called weak coercivity because of the complex magnitude used in the left-hand side of these inequalities. Theorem For every choice of ω, h, γ 0, γ 1 > 0, f L Ω, and g L Ω there exists a unique solution u h of 3.6. Proof. This is an immediate consequence of Theorem and the well-known Lax- Milgram-Babuška theorem [8, 9]. We note that the weak coercivity of A h, in 3.1 depends in an adverse way on the mesh parameter h. For this reason, this weak coercivity cannot be used to obtain optimal error estimates in the case that h is allowed to be arbitrarily small. In the case of small h, which belongs to the asymptotic mesh regime, we instead rely on a Gärding s inequality for A h, to derive more robust estimates. This Gärding s inequality is proved in the following theorem: Theorem For v h V h the following Gärding s inequality for A h, holds: 1 v h 1,h ω ρ v h L Ω Cξ A hv h, v h, 3.16 where ξ := 1 + 1γ0 and C is independent of ω, h, γ 0, γ 1. Proof. Similar to the proof of the weak coercivity inequalities in Theorem 3.1.3, we begin by taking the real and imaginary part of A h, separately c.f Also, in a similar fashion as was done in the proofs of Theorem 3.1. and 3.1.3, the Cauchy-Schwarz and Young s inequality along with 3.7 and 3.8 are used to bound the term e Eh I {σvh n e }, [v h ]. Thus the following sequence of inequalities are e 61

78 obtained: Re A h v h, v h = v h 1,h ω ρ v L Ω + Re e E I h v h 1,h ω ρ v h L Ω 1 v h 1,h 1 v h 1,h ω ρ v h L Ω {σvh n e }, [v h ] e Cλ + µ γ 0 J 0 v h, v h Cλ + µ γ 0 Im A h v h, v h. Rearranging the above inequality and adding 1 times 3.15 yield 1 v h 1,h ω ρ v h L Ω C Re A hv h, v h + C C 1 + λ + µ γ 0 Hence, 3.16 holds. 1 + λ + µ Im A h v h, v h γ 0 A h v h, v h. 3. Asymptotic Error Estimates Recall that Theorem guarantees both continuity and weak coercivity of A h, for any positive values of ω, h, γ 0, γ 1. Unfortunately, as is observed in Theorem 3.1.3, the weak coercivity inequality degrades as h becomes small. For this reason, weak coercivity of A h, cannot be used to obtain optimal order error estimates in the asymptotic mesh regime, i.e. ω h = O1. Instead, a standard argument called Schatz argument c.f. [71] is often used to obtain error estimates in the asymptotic mesh regime. Schatz argument is useful for deriving error estimates for consistent discretizations of indefinite problems that are characterized by sesquilinear forms that satisfy a Gärding s inequality. This method has been used in the past to prove optimal order error estimates for finite element approximations of Helmholtz-type problems. For references of Schatz argument applied to finite element formulations of the scalar Helmholtz equation, see [6, 9, 30]. For an example of Schatz argument applied to a finite element approximation of the 6

79 elastic Helmholtz equation, see [6]. For a general reference of Schatz argument applied to the finite element approximations of general second order PDEs satisfying Gärding s inequality, see [16]. The consistency of 3.6 immediately infers Galerkin orthogonality on V h. Namely, A h u u h, v h = 0 v h V h, 3.17 where u solves and u h solves 3.6. We also quote frequency-explicit a priori estimates for the PDE solution u. These are taken from [7]. Note that the estimates in [7] were only carried out in the case in which g = 0 but these estimates should hold as well for any g L Ω. This extension can be expected because the analysis in Theorem.5. holds when g L Ω and this analysis is based on that of [7]. The estimate that will be used in the analysis of the proposed IP-DG method is quoted below. Theorem Suppose that Ω is a convex polygonal domain or Ω is a smooth domain. Further suppose that u H Ω solves Then u satisfies the following regularity estimate: u H Ω C ω α + 1 ω f L Ω + g L Ω, 3.18 where α = 1 if u V K for some positive constant K as defined in Chapter and otherwise α = Elliptic Projection and its Error Estimates The primary goal of this section is to estimate the error u u h in the asymptotic mesh regime. This will be done based on the error decomposition u u h = u ũ h + ũ h u h with some ũ h V h which is sufficiently close to u, such as a projection of u. To this end, for any w E, we define its elliptic projection w h V h as the 63

80 solution to the following problem: a h w h, v h + iω A w h, v h Ω = a hw, v h + iω Aw, v h Ω v h V h Since the elliptic projection is defined using the bilinear form a h,, it would be prudent to prove some properties of a h,. The following theorem is given with this in mind. Theorem 3... For any v, w E there exists a positive constant C independent of ω, h, γ 0, γ 1 such that a h v, w C v 1,h w 1,h. 3.0 Also for any 0 < δ < 1 and v h V h there holds Re a h v h, v h + 1 δ + C δ Im a h v h, v h 1 δ v h γ 1,h Proof. Note that 3.0 is easy to prove with the techniques used to prove continuity of A h, and thus we omit it. By using Cauchy-Schwarz, trace, inverse, and Young s inequalities we get Re a h v h, v h v h 1,h + Re e E I h [vh ], {σv h n e } e v h 1,h e E I h v h 1,h e E I h [v h ] L e {σv h n e } L e 1 0 C γ [v γ 1 0 h 1 h ] L e e 1 δ v h 1,h C δ γ 0 J 0 v h, v h. σvh L K e + σv h L K e 64

81 Also, Im a h v h, v h = J 0 v h, v h + J 1 v h, v h. Combining these two results yields 3.1. Using this theorem, it is easy to check that the bilinear form a h, + iω A, Ω is both continuous and coercive when γ 0 is large enough. Hence, by the Lax-Milgram theorem, the above elliptic projection is well-defined. Trivially, the following Galerkin orthogonality holds. Lemma Suppose that w E and w h V h is its elliptic projection then a h w w h, v h + iω Aw w h, v h = 0 vh V h. 3. Theorem Let u H Ω solve and let ũ h V h be its elliptic projection defined in Then the following estimates hold: u ũ h 1,h + ω 1 ξ u ũh L Ω 3.3 Cξ h ξ + γ 1 + ωh 1 ω α + 1 f L ω Ω + g L Ω, 3.4 and u ũ h L Ω Cξ h ξ + γ 1 + ωh ω α + 1 ω f L Ω + g L Ω, 3.5 where ξ = 1 + γ 1 0, α is defined in Theorem 3..1, and C is a positive constant independent of ω, h, γ 0, γ 1. Proof. Let û h V h denote the P 1 conforming finite element interpolant of u on T h. Then the following estimates are well-known c.f. [16, 4]: u û h L Ω Ch u H Ω and u û h L Ω Ch u H Ω

82 Applying the trace and inverse inequalities to these estimates yields u û h 1,h Cξ + γ 1 1 h u H Ω, 3.7 and u û h L Ω Ch 3 u H Ω. 3.8 Set ψ h := ũ h û h. By Galerkin orthogonality along with the fact that ψ h +u ũ h = u û h we get a h ψ h, ψ h + iω Aψ h, ψ = a h Ω hu û h, ψ h + iω Au û h, ψ h Ω. 66

83 Next, it follows from Theorem 3.. with δ = 1 and Lemma that 1 ψ h 1,h Cξ ψ h 1,h Cξ Re a h ψ h, ψ h + Cξ 1 + C 1 γ 0 Im a h ψ h, ψ h = Cξ Re a h ψ h, ψ h + iω Aψ h, ψ h Cξω Ω C 1 Aψh, ψ h + Cξ + C 1 Im a h ψ γ h, ψ h + iω Aψ h, ψ h Ω 0 = Cξ Re a h u û h, ψ h + iω Au û h, ψ h Ω 1 + Cξ + C 1 Im a h u û h, ψ γ h + iω Au û h, ψ h Ω 0 1 Cξω + C 1 Aψh, ψ h Ω γ 0 γ 0 Ω Cξ ψ h 1,h u û 1,h + Cωξ ψ h L Ω u û L Ω Cωξ ψ h L Ω + Cξ ψ h 1,h u û 1,h + ω ψ h L Ω u û L Ω Cξ ψ h 1,h u û 1,h + Cωξ u û h L Ω C 4 ωξ ψ h L Ω Cξ 4 u û h 1,h ψ h 1,h C 4 ωξ ψ h L Ω + Cωξ u û h L Ω. Substituting 3.7 and 3.8 into the above estimate gives ψ h 1,h + ωξ ψ h L Ω ξ C 4 u û h 1,h + ωξ u û h L Ω C ξ 4 ξ + γ 1 h u H Ω + ωξ h 3 u H Ω = Cξ 4 h ξ + γ 1 + ωh u H Ω Cξ 4 h ξ + γ 1 + ωh ω α + 1 ω f L Ω + g L Ω. 67

84 Thus, ψ h 1,h + ω 1 ξ ψh L Ω Cξ h ξ + γ 1 + ωh 1 ω α + 1 ω f L Ω + g L Ω. Recall that u ũ h = u û h ψ h. By the triangle inequality we get u ũ h 1,h + ω 1 ξ u ũh L Ω u û h 1,h + ω 1 ξ u ûh L Ω + ψ h 1,h + ω 1 ξ ψh L Ω Cξ h ξ + γ 1 + ωh 1 ω α + 1 f L ω Ω + g L Ω. Therefore, 3.4 holds. To obtain 3.5, we appeal to the duality argument by considering the following auxiliary PDE problem: div σw = u ũ h in Ω, σwn iωaw = 0 on Ω. It can be shown that there exists a unique solution w H Ω such that w H Ω C u ũ h L Ω

85 Let ŵ h V h be the P 1 conforming finite element interpolant of w. Testing the above auxiliary problem with u ũ h yields u ũ h L Ω = u ũ h, div σw Ω = a h u ũ h, w + iω Au ũ h, w Ω = a h u ũ h, w ŵ h + iω Au ũ h, w ŵ h Ω C u ũ h 1,h w ŵ h 1,h + ω u ũ h L Ω w ŵ h L Ω C ξ + γ 1 1 h w H Ω u ũ h 1,h + ωh 3 w H Ω u ũ h L Ω C ξ + γ 1 1 h u ũh L Ω u ũ h 1,h + ωh 3 u ũh L Ω u ũ h L Ω. Hence, u ũ h L Ω Chξ + γ 1 + ωh 1 u ũ h 1,h + ω 1 ξ u ũh L Ω Cξ h ξ + γ 1 + ωh ω α + 1 f L ω ω + g L Ω. Thus, 3.5 holds. With estimates for the elliptic projection in hand, all of the components are in place to complete Schatz argument and obtain asymptotic error estimates. The next subsection is devoted to carrying this out. 3.. Asymptotic Error Estimates Via Schatz Argument In this subsection, error estimates for the proposed IP-DG method are proved in the asymptotic mesh regime i.e. when ω h = O1. This will be carried out using the well-known Schatz argument c.f. [6, 9, 30, 6, 16]. 69

86 To carry out Schatz argument, we introduce the error decomposition e h := u u h = ψ + φ h, where ψ := u ũ h, φ h := ũ h u h, and ũ h is the elliptic projection of u as defined in Recall that estimates on ψ have already been established c.f. Theorem Thus it is left to bound φ h. The next step relates the semi-norm φ h 1,h to norms on ψ and a lower order norm φ h L Ω. This step relies on the Gärding s inequality for A h, and the continuity of A h,. After this is complete, the next step is to relate the norm e h L Ω to the semi-norm h e h 1,h using a duality argument. The final step is to put all of the previous steps together with h chosen to be small enough to obtain the desired error estimates. The following lemmas carry out the intermediate steps leading to error estimates in the asymptotic mesh regime. Lemma Suppose that u H Ω solves , u h is its IP-DG approximation, and ũ h is its elliptic projection. Then for φ h = ũ h u h φ h 1,h C ξ 3 ψ 1,h + ξ 3 ω ρ ψ L Ω + ω ρ φ h L Ω, 3.30 where ψ = u ũ h, ξ = γ 0 and C is a positive constant independent of ω, h, γ 0, γ 1. Proof. By Gärding s inequality c.f. Theorem 3.1.6, the continuity of A h,, Galerkin orthogonality, the Cauchy-Schwarz and Young s inequalities we have 1 φ h 1,h ω ρ φ h L Ω Cξ A hφ h, φ h = Cξ A h ψ, φ h 1 1 Cξ ψ 1,h + ω ρ ψ L Ω ξ φ h 1,h + ω ρ φ h L Ω C ξ 3 ψ 1,h + ξ 3 ω ρ ψ L Ω + ω ρ ξ φ h L Ω φ h 1,h. 70

87 Thus, rearranging the terms and using the fact that ξ inequality Lemma Suppose that u H Ω solves and u h yields the desired is its IP-DG approximation. There exists a positive constant C 1 independent of ω, h, γ 0, γ 1 such that for h min { 1 ω, C 1 ξ ωξ + γ 1 ω α + 1ω } 1, there holds u u h L Ω Cξ h ξ + γ 1 ω α + 1ω u u h 1,h, 3.31 where ξ = γ 0 and C is a positive constant independent of ω, h, γ 0, γ 1. Proof. Let w be the solution to the following problem: A h v, w = v, e h Ω v E. Let w h be the elliptic projection of w defined by A h, and Galerkin orthogonality we get Using the continuity of e h L Ω = e h, e h L Ω = A h e h, w = A h e h, w w h 1 1 C e h 1,h + ω e h L Ω w w h 1,h + ω w w h L Ω. 71

88 Applying Theorem 3..4 and using the fact that h 1 ω yield e h L Ω C e h 1,h + ω e h L Ω 1 ξ ξ + γ 1 + ωhh1 + ωh C 1 e h 1,h + ω e h L Ω ξ hξ + γ 1 1 ω α + 1ω e h L Ω. ω α + 1ω e h L Ω Dividing both sides of this inequality by e h L Ω and using the fact that h C1 ξ ωξ + γ 1 ω α ω gives e h L Ω C 1 ξ hξ + γ 1 ω α + 1ω e h 1,h + C 1 ξ ωhξ + γ 1 ω α + 1ω e h L Ω C 1 ξ hξ + γ 1 ω α + 1ω e h 1,h + 1 e h L Ω. Subtracting 1 e h L Ω result. from both sides of the above inequality yields the desired Lemma 3..5 and lemma 3..6 are now put together to derive optimal order error estimates for h small. Theorem Let e h = u u h where u H Ω solves and u h is its IP-DG approximation. There { exists a positive constant C, independent of ω, h, γ 0, γ 1, 1 such that for h h 0 := min, Cξ 5 ωξ + γ ω 1 } ω α the following error ω 7

89 estimates hold: e h 1,h Cξ 4 ξ + γ 1 ω α + 1 h + ωh f ω L Ω + g L Ω, 3.3 e h L Ω Cξ 6 ξ + γ 1 ω α + 1 ω h + ωh 3 f L Ω + g L Ω, 3.33 where ξ = γ 0, α is defined as in Theorem 3..1, and C is a positive constant independent of ω, h, γ 0, γ 1. Proof. Let ũ h be the elliptic projection of u. Then e h = ψ + φ h, where ψ = u ũ h and φ h = ũ h u h. By Lemma 3..5 we get e h 1,h C ψ 1,h + φ h 1,h C ψ 1,h + ξ φ h 1,h C ξ 4 ψ 1,h + ξ 4 ω ψ L Ω + ξω φ h L Ω Note that φ h = ψ + e h, and the triangle inequality implies that φ h L Ω C ψ L Ω + e h L Ω. Substituting the above inequality into 3.34 and using Lemma 3..6 at this point we assume that C C 1 and thus h is small enough to satisfy the condition of Lemma 3..6, we have e h 1,h C ξ 4 ψ 1,h + ξ 4 ω ψ L Ω + ξω e h L Ω C ξ 4 ψ 1,h + ξ 4 ω ψ L Ω + Cξ 5 ω h ξ + γ 1 ω α + 1 ω e h 1,h. 73

90 Now the fact that h h 0 implies Thus, e h 1,h C ξ 4 ψ 1,h + ξ 4 ω ψ 1,h e h 1,h. e h 1,h C ξ ψ 1,h + ξ ω ψ L Ω, which together with Theorem 3..4 infers follows from applying Lemma 3..6 to 3.3. We note that a mesh constraint of the form ω α+1 h = O1 must be used to ensure an optimal order error estimate when using Schatz argument. If a Korn s inequality holds on the boundary c.f. Conjecture.3.4 then α = 1 and the constraint becomes ω h = O1. This is consistent with the mesh constraint used to characterize the asymptotic mesh regime for discretization methods applied to the other Helmholtz-type problems. Stability in the asymptotic mesh regime can be derived as a consequence of the error estimates above. For standard discretization methods applied to the elastic Helmholtz problem, stability has only been proved in the asymptotic mesh regime. In the next section, we will carry out a stability and convergence analysis for our IP-DG method in the pre-asymptotic mesh regime, i.e. for ω α+1 h > C. 3.3 Pre-asymptotic Error Estimates In the previous section, optimal error estimates were obtained in the asymptotic mesh regime, i.e. when ω α+1 h C, where α is defined in Theorem 3..1 and C is some positive constant independent of ω. In this section, stability estimates for our IP- DG approximation will be established in the pre-asymptotic mesh regime, i.e. when ω α+1 h > C. These stability estimates will then be used to establish optimal order 74

91 error estimates in the pre-asymptotic regime. Thus, Section 3. together with this section ensure that the proposed IP-DG method is absolutely stable. This property makes the proposed IP-DG method especially well suited to approximate the elastic Helmholtz equations Stability Estimates in the Pre-Asymptotic Mesh Regime In this subsection, stability estimates for the proposed IP-DG method are established. Specifically, these estimates are established in the pre-asymptotic mesh regime, i.e. when ω α+1 h > C. It turns out that stability in this case is just a consequence of Theorem which proves a weak coercivity property of A h,. As stated previously, the coercivity constant from this theorem is adversely dependent on h for small values of h. In the case that h is bounded away from zero, this constant can be replaced with one that is not dependent on h. Thus, stability estimates for the pre-asymptotic mesh regime are obtained as a consequence of the weak coercivity of A h,. The stability of the IP-DG method in the pre-asymptotic mesh regime is given by the next theorem. Theorem Suppose that u h V h solves the IP-DG method given by 3.6. Then the following inequalities hold: J 0 u h, u h + J 1 u h, u h + Au h, u h where C sta = of ω, h, γ 0, γ 1. u h 1,h + ω ρ u h L Ω C C sta f L Ω + C sta g L Ω C sta f L Ω + g L Ω Ω C ω, 3.35, 3.36 ξ ω + 1 ω h + 1 ω 3 h γ 1, ξ = γ 0, and C is a positive constant independent 75

92 Proof. By 3.1 and 3.13 we get u h 1,h + ω ρ u h L Ω + ω γ C γ 0 ωh + 1 ω ρh γ 1 C γ 0 ωh + 1 Auh, u ω ρh h γ Ω 1 A h u h, u h ωh + 1 f ω ρh L γ Ω u h L Ω + g L Ω u h L Ω 1 C ω ρ γ 0 ωh + 1 f ω ρh L γ Ω + ω ρ 1 u h L Ω + C ωc A γ ωh + 1 ω ρh γ 1 g L Ω + ωc A γ ωh + 1 ω ρh γ 1 u h L Ω. Substituting 3.10 into the above inequality infers Now, combining 3.13 with 3.35 yields J 0 u h, u h + J 1 u h, u h + ω Au h, u h Ω A h u h, u h f L Ω u h L Ω + g L Ω u h L Ω C 1 ρ C sta f L Ω + 1 C sta g L c Ω A f ωρ 1 L Ω 1 + C g L ωc Ω + ωc A A u h L Ω C ωρ C sta f L Ω + C g L ωc Ω + ωc A A u h L Ω. Using 3.10 in the above inequality infers Remark When ω α+1 h > C ξ C sta < C ω + ωα 1 + ωα γ 1 Thus, the constant in the above stability estimate is independent of h in the preasymptotic mesh regime. 76

93 3.3. Error Estimates for the IP-DG Method In this subsection, the stability estimates established in Subsection are utilized to obtain optimal order error estimates for the proposed IP-DG method in the preasymptotic mesh regime i.e. ω α+1 h > C. These estimates are obtained with the help of the elliptic projection defined in Subsection 3..1, the stability estimates established in Subsection 3.3.1, and Galerkin orthogonality for the sesquilinear form A h,. Let u H Ω solve and u h V h be its IP-DG approximation defined in 3.6. As before, the error e h is defined by e h := u u h. Subtracting 3.6 from 3.3 immediately yields the following Galerkin orthogonality property for A h, : a h e h, v h ω ρ e h, v h + iω Ae Ω h, v h Ω = 0 v h V h Let ũ h V h denote the elliptic projection of u as defined in In the same fashion as was done in subsection 3.., the error e h can be decomposed as e h = ψ+φ h where ψ = u ũ h and φ h = ũ h u h. Again, estimates on e h will be established from estimates on ψ and φ h that are obtained separately. By Galerkin orthogonality given in 3.17 and 3.38 we have the following identity: a h φ h, v h ω ρ φ h, v h + iω Aφ Ω h, v h 3.39 Ω = a h ψ, v h + ω ρ ψ, v h iω Aψ, v Ω h Ω = ω ρ ψ, v h Ω v h V h. In other words, φ h V h solves 3.6 with f = ω ρψ and g 0. This allows us to establish estimates on φ h by using the stability estimates from Theorem Specifically, we have the next lemma. 77

94 Lemma Let u H Ω solve , u h be its IP-DG approximation, and ũ h be its elliptic projection. Then φ h = ũ h u h satisfies the following inequality: φ h 1,h + ωρ 1 φh L Ω where C sta = of ω, h, γ 0, γ 1. Cξ ω h C sta ξ + γ1 + ωh ω α + 1 ω f L Ω + g L Ω, 3.40 ξ ω + 1 ω h + 1 ω 3 h γ 1, ξ = γ 0, and C is a positive constant independent Proof implies that φ h solves 3.6 with f = ω ρψ and g 0. Thus, an application of Theorem yields φ h 1,h + ωρ 1 φh L Ω C ρ 1 C sta ω ρψ L Ω Cω ρ 1 Csta ψ L Ω, which along with Theorem 3..4 infers We are now ready to derive error estimates for our IP-DG method in the preasymptotic mesh regime. The next theorem is a consequence of combining Theorem 3..4 and Lemma Theorem Let u H Ω solve and u h be its IP-DG approximation. Then e h = u u h satisfies the following inequality: e h 1,h + ωρ 1 eh L Ω 3.41 Cξ h ξ + γ 1 + ωh ω α + 1 f ω L Ω + g L Ω + Cξ ωh 1 + ωc sta ξ + γ1 + ωh ω α + 1 f ω L Ω + g L Ω, e h L Ω Cξ h 1 + ωc sta ξ + γ1 + ωh ω α + 1 ω f L Ω + g L Ω,

95 where C sta = of ω, h, γ 0, γ 1. ξ ω + 1 ω h + 1 ω 3 h γ 1, ξ = γ 0, and C is a positive constant independent Proof. Recall that estimates for ψ and φ h have already been established in Theorem 3..4 and Lemma 3.3.3, respectively. These estimates are combined in the following steps to obtain 3.41: e h 1,h + ωρ 1 eh L Ω ψ 1,h + ωρ 1 ψ L Ω + φ h 1,h + ωρ 1 φh L Ω Cξ h ξ + γ 1 + ωh 1 ω α f ω L Ω + g L Ω + Cξ ωh ξ + γ 1 + ωh ω α + 1 ω f L Ω + g L Ω + Cξ ω h C sta ξ + γ1 + ωh ω α + 1 f ω L Ω + g L Ω Cξ h ξ + γ 1 + ωh ω α + 1 f ω L Ω + g L Ω + Cξ ωh 1 + ωc sta ξ + γ1 + ωh ω α + 1 ω f L Ω + g L Ω. Similarly, 3.4 is obtained by combining Theorem 3..4 and Lemma Remark When ω α+1 h > C ξ C sta < C ω + ωα 1 + ωα 1. γ 1 Therefore, the above error estimates are optimal in h in the pre-asymptotic mesh regime. 3.4 Numerical Experiments In this section, numerical tests are carried out in order to demonstrate key features of the proposed IP-DG method. We choose Ω = 0.5, 0.5 R i.e. the unit square in R centered at the origin, along with the material constants ρ = µ = λ = 1, 79

96 and penalty constants γ 0 = 10 and γ 1 = 0.1. For the sake of testing the exact error, f and g are chosen so that the exact solution to the elastic Helmholtz equations is u = 1 ω r [eiωr 1, e iωr 1] T, where r = x. This simple problem along with the subsequent numerical tests are chosen to mirror those for the IP-DG method proposed in [4] for the scalar Helmholtz problem. Some example plots are given in Figures 3. and 3.3. These plots demonstrate how well the proposed IP-DG method can capture an example with large wave frequency when using a relatively coarse mesh. To partition the domain Ω, a uniform triangulation T h is used. For a positive integer n, define T 1/n to be a triangulation of n congruent isosceles triangles with side lengths 1/n, 1/n, and /n. Figure 4.1 shows a sample triangulation T 1/10. The numerical tests in this section intend to demonstrate the following: absolute stability of our IP-DG method, error of our IP-DG solution, pollution effect on the error when ωh = O1, absence of the pollution effect when ω 3 h = O1, comparisons between standard FE and our IP-DG method for this problem. Figure 3.1: Example of the triangulation T 1/10. 80

97 Figure 3.: Plot of Re u h for ω = 50 and h = 1/70. Both a top down view left and a side view right are shown. Figure 3.3: Plot of Re u h for ω = 100 and h = 1/10. Both a top down view left and a side view right are shown Stability In this subsection, the stability of both the proposed IP-DG method and the P 1 - conforming finite element method will be discussed. conforming finite element approximation of u. Let u F EM h denote the P 1 - Recall that the proposed IP-DG approximation is absolutely stable, i.e. it is stable for all ω, h, γ 0, γ 1 > 0. This has not been established for the P 1 -conforming finite element approximation. In fact, the stability of the P 1 -conforming finite element approximation is known to hold only when h satisfies ω h C. 81

98 Figure 3.4 plots both u h 1,h and u F EM 1,h for h = 0.05, 0.01 and ω = 1,,..., 00. We observe that u h 1,h decreases in a smooth fashion as ω increases. This smooth behavior of u h 1,h is indicative of the absolute stability of the IP-DG approximation. On the other hand, we observe oscillations in u F h EM 1,h that occur when we vary ω. This oscillation is indicative of the instability of the P 1 -conforming finite element method when h is too large. Figure 3.4: Plots of u h 1,h and u F EM h 1,h Error In this subsection, the optimal order of convergence for the proposed IP-DG method will be demonstrated. The pollution effect will also be demonstrated. From Theorems and 3..7 we expect the error in 1,h to decrease at an optimal order in both the pre-asymptotic and asymptotic mesh regimes. In other words, u u h 1,h = Oh is expected. Figure 3.5 is a log-log plot of the relative error u u h 1,h / u 1,h against the value 1/h for frequencies ω = 5, 10, 0, 30. From this plot, it is observed that the relative error decreases at the same rate as h, thus displaying the optimal order of convergence in the relative semi-norm. Also displayed in Figure 3.5 is the error when ω varies according to the constraint ωh = 0.5. From this figure it is observed that the error increases as ω increases under this constraint. This is due to the pollution effect on the error for the elastic Helmholtz equations. 8

99 Figure 3.5: Log-log plot of the relative error for the IP-DG approximation measured in the H 1 -seminorm for different values of ω. The pollution effect for Helmholtz-type problems is characterized by the increase in error as ω is increased under the constraint ωh = O1. This effect is intrinsic to Helmholtz-type problems c.f. [54]. It is well-known that the pollution effect can be eliminated if h is chosen to fulfill the more stringent constraint ω 3 h = O1. In Figure 3.6 the relative error is plotted against ω as h is chosen to satisfy different constraints. Under the constraints ωh = 1 and ωh = 0.5, the pollution effect is present and the relative error increases as ω is increased. On the other hand, when ω 3 h = 1 is used to choose the the mesh size h, the pollution effect is eliminated. Figure 3.6: Relative error of the IP-DG approximation measured in the H 1 seminorm computed for different values of ω and h is chosen to satisfy the given constraints. 83

100 3.4.3 IP-DG vs. FEM In this subsection, the proposed IP-DG solution is compared to the P 1 -conforming finite element solution. As stated previously, the proposed IP-DG method is absolutely stable while the P 1 -conforming finite element method is only shown to be stable when h satisfies ω h = O1. With this in mind, one can anticipate that in the case that the frequency ω is allowed to be large, the IP-DG method becomes a better method. In Figures , Reu h and Reu F h EM are plotted for ω = 100 and h = 1/50, 1/10, 1/00 on a cross-section over the line y = x. In addition, Reu is plotted to measure how well the respective approximations capture the true solution. In Figure 3.7, it is observed that u h already captures the phase of u with h = 1/50 while not fully capturing the large changes in magnitude. On the other hand, for h = 1/50, u F EM h has spurious oscillations. In this case, u F EM h also fails to capture the changes in the magnitude of the wave. In Figure 3.8, we see that for h = 1/10, u h captures the phase and changes in magnitude of the wave very well while u F EM h still displays spurious oscillations. In Figure 3.9, we see for h = 1/00, both methods capture the wave well. However, the IP-DG method captures the wave slightly better. These examples demonstrate that the IP-DG method approximates high frequency waves better than the standard finite element when a coarse mesh is employed. This is of great importance when memory is limited or one wishes to employ a multi-level solver such as multigrid or multi-level Schwarz space/domain decomposition methods. 84

101 Figure 3.7: The left plot is of Reu h solid red line vs. Reu dashed blue line for h = 1/50. The right plot is of Reu F h EM solid red line vs. Reu dashed blue line for h = 1/50. Figure 3.8: The left plot is of Reu h solid red line vs. Reu dashed blue line for h = 1/10. The right plot is of Reu F h EM solid red line vs. Reu dashed blue line for h = 1/10. 85

102 Figure 3.9: The left plot is of Reu h solid red line vs. Reu dashed blue line for h = 1/00. The right plot is of Reu F h EM solid red line vs. Reu dashed blue line for h = 1/00. 86

103 Chapter 4 A Multi-modes Monte Carlo Interior Penalty Discontinuous Galerkin Method for Acoustic Wave Scattering in Random Media 4.1 Introduction Partial differential equations with random coefficients arise naturally in the modeling of many physical phenomena. This is due to the fact that some level of uncertainty is usually involved if the knowledge of the physical behavior or when noise is present in the experimental measurements. In recent years, substantial progress has been made in the numerical approximation of such PDEs due to the significant development in computational resources. We refer to [11, 1, 1, 68, 80] and the references therein for more details. In this chapter, we consider the propagation of acoustic waves in a medium where the wave velocity is characterized by a random process. More precisely, we study the 87

104 approximation of the solution to the following Helmholtz problem: uω, k αω, uω, = fω, in D, 4.1 ν uω, + ikαω, uω, = 0 on D, 4. where k is the wave number, and D R d d = 1,, 3 is a convex bounded polygonal domain with boundary D. Let Ω, F, P be a probability space with sample space Ω, σ algebra F and probability measure P. For each fixed x D, the refractive index α, x is a real-valued random variable defined over Ω. We assume that the medium is a small random perturbation of a uniform background medium in the sense that αω, := 1 + εηω,. 4.3 Here ε represents the magnitude of the random fluctuation, and η L Ω, L D is some random process satisfying } P {ω Ω; ηω, L D 1 = 1. For notational brevity, we only consider the case that η is real-valued. However, we note that the results of this chapter are also valid for complex-valued η. On the boundary D, an absorbing boundary condition is imposed to absorb incoming waves [35]. Here ν denotes the unit outward normal to D, and ν u stands for the normal derivative of u. The boundary value problem arises in the modeling of wave propagation in complex environments, such as composite materials, oil reservoirs and geological basins [46, 55]. In such instances, it is of practical interest to characterize the uncertainty of the wave energy transport when the medium contains some randomness. In particular, we are interested in the computation of some statistics of the wave field, e.g. the expected value of the solution u. To solve stochastic or random partial differential equations SPDEs numerically, the simplest and most natural approach is to use the Monte Carlo method, where 88

105 a set of independent identically distributed i.i.d. solutions are obtained by sampling the PDE coefficients, and the expected value of the solution is calculated via a statistical average over all the sampling in the probability space [1]. An alternative is the stochastic Galerkin method, where the SPDE is reduced to a high dimensional deterministic equation by expanding the random field in the equation using the Karhunen-Loève or Wiener Chaos expansions. We refer the reader to [11, 1, 33, 68, 80] for detailed discussions. However, it is known that a bruteforce Monte Carlo or stochastic Galerkin method applied directly to the Helmholtz equation with random coefficients is computationally prohibitive even for a moderate wave number k, since a large number of degrees of freedom is involved in the spatial discretization. It is apparent that in such cases, the Monte Carlo method requires solving a PDE with many sampled coefficients, while the high dimensional deterministic equation associated with the stochastic Galerkin method will be too expensive to be solved. In this chapter, we propose an efficient numerical method for solving the Helmholtz problem when the medium is weakly random defined by 4.3. A multimodes representation of the solution is derived, where each mode is governed by a Helmholtz equation with deterministic coefficients and a random source. We develop a Monte Carlo interior penalty discontinuous Galerkin MCIP-DG method for approximating the mode functions. In particular, we take advantage of the fact that the coefficients of the Helmholtz equation for all the modes are identical; hence, the associated discretized equations share the same constant coefficient matrix. Using this crucial fact, it is observed that an LU direct solver for the discretized equations leads to a tremendous saving in the computational costs, since the LU decomposition matrices can be used repeatedly. Thus, all of the solutions for all modes and all samples can be obtained in an efficient way by performing simple forward and backward substitutions. Indeed, it turns out that the computational complexity of the proposed algorithm is comparable to that of solving a few deterministic Helmholtz problems using the LU direct solver. 89

106 The rest of the chapter is organized as follows. A wave-number explicit estimate for the solution of the random Helmholtz equation is established in Section 4.. In Section 4.3, we introduce the multi-modes expansion of the solution as a power series of ε and analyze the error estimation for its finite-modes approximation. The Monte Carlo interior penalty discontinuous Galerkin method is presented in Section 4.4, where the error estimates for the approximation of each mode function is also obtained. In Section 4.5, a numerical procedure for solving is described and its computational complexity is analyzed in detail. In addition, we derive optimal order error estimates for the proposed procedure. Several numerical experiments are provided in Section 6 to demonstrate the efficiency of the method and to validate the theoretical results. 4. PDE Analysis The focus of this section will be to derive a priori solution estimates for the random Helmholtz problem introduced in These a priori estimates will be used to prove existence and uniqueness of the solution to the random Helmholtz problem. The techniques in this chapter will mirror those carried out for the deterministic scalar Helmholtz problem c.f. Chapter and [7] Preliminaries Similar to Chapter, analysis in this section will be carried out using the following special function spaces: H 1 +D := { v H 1 D; v Γ L D }, 4.4 V := { v H 1 D; v L D }. 4.5 Without loss of generality, we assume that the domain D B R 0. Throughout this chapter we also assume that D is a star-shaped domain with respect to the origin 90

107 in the sense that there exists a positive constant c 0 such that x ν c 0 on D, Recall that the analysis in Chapter also relied on a star-shape condition on the domain. Let Ω, F, P be a probability space on which all the random variables of this chapter are defined. E denotes the expectation operator on this probability space. The abbreviation a.s. stands for almost surely. As it will be needed in the late sections of this chapter, in this section we analyze the boundary value problem for the Helmholtz equation 4.1 with the following slightly more general nonhomogeneous boundary condition: ν uω, + ikαω, uω, = gω,. 4.6 As in Chapter, analysis of the random Helmholtz problem 4.1,4.6 will be carried out on its weak formulation. With this in mind, we introduce the following definition. Definition Let f L Ω, L D and g L Ω, L D. A function u L Ω, H 1 D is called a weak solution to problem 4.1,4.6 if it satisfies the following identity: au, v dp = f, vd + g, v D dp v L Ω, H 1 D, 4.7 where Ω Ω aw, v := w, v D k α w, v D + ik αw, v D. 4.8 Remark 4... Using 4.10 below, it is easy to show that any solution u of 4.1,4.6 satisfies u L Ω, H 1 +D V. 91

108 4.. Wave-number Explicit Solution Estimates In this subsection, we shall derive stability estimates for the solution of problem 4.1,4.6 which is defined in Definition Similar to Chapter, our focus is to obtain explicit dependence of the stability constants on the wave number k. Such wave-number explicit stability estimates will play a vital role in our convergence analysis in the later sections. We note that wave-number explicit stability estimates also play a pivotal role in the development of numerical methods, such as finite element and discontinuous Galerkin methods, for deterministic reduced wave equations cf. Chapter 3 and [4, 44]. As a byproduct of the stability estimates, the existence and uniqueness of solutions to problem 4.1,4.6 will be conveniently established. Lemma Let u L Ω, H 1 D be a solution of 4.1,4.6, then for any δ 1, δ > 0 and ε < 1 there hold E u L D k 1 + ε + δ 1 E u L D 4.9 δ 1 + k 1 ε + 1 E f L δ D + E g L D, 1 E u L D δ k1 ε E u L D + 1 δ k1 ε E f L D k 1 ε E g L D. Proof. Setting v = u in 4.7 yields Ω au, u dp = Ω f, ud + g, v D dp. Taking the real and imaginary parts and using the definition of a,, we get Ω u L D k 1 + εηu L D dp = Re k 1 + εη, u D dp = Im Ω Ω Ω f, ud + g, v D dp, 4.11 f, ud + g, v D dp

109 Applying the Cauchy-Schwarz inequality to 4.1 produces k1 εe u L D δ E u L D + 1 δ E f L D + k1 ε E u L D + 1 k1 ε E g L D. Thus, 4.10 holds. Applying Cauchy-Schwarz to 4.11 yields E u L D k 1 + ε + δ 1 E u L D + 1 E f L δ D 1 + δ 1 E u L D + 1 E g L δ D, 1 which together with 4.10 using δ = k1 ε infers 4.9. The proof is complete. From Lemma 4..3, We observe that the test function v = u is not enough to obtain solution estimates for the weak solution of 4.1, 4.6. Recall that this was also the case for the deterministic scalar Helmholtz equation. To overcome this difficulty, in Chapter and [7] we made use of Rellich identities for the Laplacian operator. With this in mind, we prove the following lemma. Lemma Let u L Ω, H D, then Re Ω Re u, x u dp = d D Ω u, x u D dp = d u L D dp + 1 x ν, u D dp, 4.13 Ω Ω u L D dp 4.14 Ω + 1 x ν, u dp. D Proof. To obtain the above result, we apply Lemmas.. and..3 with α = x and integrate the subsequent identities over the probability space Ω, F, P. Remark could be called a stochastic Rellich identity for the Laplacian. Ω 93

110 We are now ready to state and prove our wave-number explicit estimate for solutions of problem 4.1, 4.6 defined in Definition Theorem Let u L Ω, H 1 +D be a solution of and R be the smallest number such that B R 0 contains the domain D. Then there hold the following estimates: E u L D + u L D + c 0 u L D E u H 1 D C 0 1 C 0 k + 1 Mf, g, 4.15 k Mf, g, 4.16 k provided that ε + ε < γ 0 := min { 1, 13 d 4d 7+5kR}. Where C0 is some positive constant independent of k and u, and Mf, g := E f L D + g L D Moreover, if g L Ω, H 1 D and u L Ω, H D, there also holds E u H D C k + 1 E f k L D + g H D Proof. To avoid some technicalities, below we only give a proof for the case u L Ω, H D. For the general case, u needs be replaced by its mollification u ρ at the beginning of the proof and followed by taking the limit ρ 0 after the integration by parts is done. A similar strategy was adopted in the proofs of the generalized weak coercivity properties in Chapter. Setting v = x u in 4.7 yields Ω u, v D k α u, v D +ik αu, v D dp = f, v D + g, v D dp Ω 94

111 Using 4.13 and 4.14 after taking the real part of 4.19 and regrouping we get dk Ω u L D dp = + Ω Ω d u L D + k ε Re η + εη, v D x ν, u k D Ω kim 1 + εηu, v D + 1 Ref, v D + Re g, v D dp. dp x ν, u D dp It then follows from the Cauchy-Schwarz inequality, the star-shape condition, and the facts that x R for x D and η L D 1 a.s. that dk 1 E u L D k εr + ε E u L δ D + δ 1 1 E u L D + d E u L D + R δ E f L D + Rδ E u L D + R δ 3 E g L D + Rδ 3 E u L D + kr δ 4 E u L D + krδ 4E u L D c 0 E u L D k R + E u L D. At this point, we note that ε + ε 1 implies ε 1. Setting δ 3 = c 0 4R, δ 4 = c 0 8kR, denoting γ = ε + ε, and using Lemma 4..3 we can bound right-hand side as 95

112 follows: c 0 + Rδ E u L D + k Rγ dk d E u L D + k Rγδ 1 8k R + + k R E u L c 0 D c 0 4 E u L D + R E f L δ D + R E g L c D 0 d + k Rγδ 1 + Rδ k 1 + γ + δ 5 E u d + + k Rγδ 1 + Rδ 8k R + + k R L D δ 1 E u L D δ5 k + 1 δ 5 E f L D + E g L D δ 6 E u L D + k δ 6 E f L D + 4 k E g L D + k Rγ E u L δ D + R E f L 1 δ D + R E g L c D 0 c 0 4 E u L D, which is equivalent to where c 1 E u L D + c 0 4 E u L D c E f L D, 4.0 c 1 := k d k k Rγδ 1 γ + δ 5 + Rδ k 1 + γ + δ 5 16k R + k R δ 6 k Rγ, c 0 δ 1 d c := + k Rγδ 1 + Rδ δ5 k + 1 δ 5 3k R + + k R R + R. k δ 6 k δ c 0 c 0 96

113 4 16R c 0 +R Let δ 1 = 1, δ k = 1, δ 4R 5 = k, and δ =, then [ 7 4d 4d 7 c 1 = k 1 + 4γRk ] + γ, d c = + krγ R 1 + c 8k 0 16R 16R R + R + 1. c 0 c 0 If γ < γ 0, it is easy to check that c 1 k. Thus, 4.0 infers that 3 E u 0,D + c 0E u L D C k k E f L D + E g L D 4.1 for some constant C > 0 independent of k and u. We combine this result with 4.10 using δ = k to obtain By 4.9 using δ 1 = k and 4.1 we get E u H 1 D = E u L D + E u L D C k k E f L D + E g L D + k 1 + ε 4 + k E u L D E f k L D + E g L D C E f L k D + E g L D. Hence, 4.16 holds. 97

114 Finally, it follows from the standard elliptic regularity theory for the Poisson equation and the trace inequality cf. [47] that E u H D E C k u + L D E f L D + E g H 1 D + E ku H 1 + D E u L D CE k 4 u L D + f L D + g H 1 D + CE k u L D + u L D C k + 1 E f k L D + g H 1. D Hence 4.18 holds. The proof is complete. Remark By the definition of γ 0, we see that γ 0 = O 1 kr. In practice, this is not a restrictive condition because R is often taken to be proportional to the wave length. Hence, kr = O1. As a non-trivial byproduct, the above stability estimates can be used conveniently to establish the existence and uniqueness of solutions to problem 4.1,4.6 as defined in Definition This strategy was mentioned at the end of Chapter and is carried out explicitly in the following theorem. Theorem Let f L Ω, L D and g L Ω, L D. For each fixed pair of positive numbers k and ε such that ε + ε < γ 0, there exists a unique solution u L Ω, H 1 +D V to problem 4.1,4.6. Proof. The proof is based on the well-known Fredholm Alternative Principle cf. [47]. First, it is easy to check that the sesquilinear form on the right-hand side of 4.7 satisfies a Gärding s inequality on the space L Ω, H 1 D. Second, to apply the Fredholm Alternative Principle we need to prove that solutions to the adjoint problem of is unique. It is easy to verify that the adjoint problem is associated 98

115 with the sesquilinear form âw, v := w, v D k α w, v D ik αw, v D, which differs from a, only in the sign of the last term. As a result, all the stability estimates for problem still hold for its adjoint problem. Since the adjoint problem is a linear problem so is problem , the stability estimates immediately infer uniqueness. Finally, the Fredholm Alternative Principle then implies that problem has a unique solution u L Ω, H 1 D. The proof is complete. Remark The uniqueness of the adjoint problem can also be shown using the classical unique continuation argument cf. [57]. 4.3 Multi-modes Representation of the Solution and its Finite Modes Approximations The first goal of this section is to develop a multi-modes representation for the solution to problem in terms of powers of the parameter ε. We first postulate such a representation and then prove its validity by establishing some energy estimates for all the mode functions. The second goal of this section is to establish an error estimate for finite modes approximations of the solution. Both the multi-modes representation and its finite modes approximations play a pivotal role in our overall solution procedure for solving problem as they provide the theoretical foundation for the solution procedure. Throughout this section, we use u ε to denote the solution to problem which is proved in Theorem We start by postulating that the solution u ε has the following multi-modes expansion: u ε = ε n u n, 4. n=0 99

116 whose validity will be justified later. Without loss of generality, we assume that k 1 and D B 1 0. Otherwise, the problem can be rescaled to this regime by a suitable change of variable. We note that the normalization D B 1 0 implies that R = 1. Substituting the above expansion into the Helmholtz equation 4.1 we get f = u ε k α u ε = ε n u n k 1 + εη + ε η u n = n=0 ε n u n k u n ε n+1 ηk u n ε n+ k η u n n=0 = u 0 k u 0 ε u 1 k u 1 k ηu 0 + ε n u n k u n k ηu n 1 k η u n. n= Matching the coefficients of ε n order terms for n = 0, 1,,, we obtain u 1 : 0, 4.3 u 0 k u 0 = f, 4.4 u n k u n = k ηu n 1 + k η u n for n Similarly, the boundary condition 4. gives 0 = uε + ik1 + εηuε ν = ε n u n ν + εn iku n + ε n+1 ikηu n n=0 = u 0 ν + iku 0 + n=1 This translates to each mode function u n as follows: ε n un ν + iku n + ikηu n 1. ν u n + iku n = ikηu n 1 for n

117 We note that the non-zero right hand side term in 4.6 was the motivation to study the random Helmholtz equation with non-homogeneous boundary data given by 4.1, 4.6 in Section 4.. A remarkable feature of the above multi-modes expansion is that all the mode functions satisfy the same type nearly deterministic Helmholtz equation and the same boundary condition. The only difference is that the Helmholtz equations have different right-hand side source terms, and each pair of consecutive mode functions supply the source term for the Helmholtz equation satisfied by the next mode function. This remarkable feature will be crucially utilized in Section 4.5 to construct our overall numerical methodology for solving problem Next, we address the existence and uniqueness of each mode function u n. Theorem Let f L Ω, L D. Then for each n 0, there exists a unique solution u n L Ω, H 1 D understood in the sense of Definition 4..1 to problem 4.4,4.6 for n = 0 and problem 4.5,4.6 for n 1. Moreover, for n 0, u n satisfies E u n L D + u n L D + c 0 u n L D 4.7 E u n H 1 D 1 k + 1 Cn, ke f k k Cn, ke f L D, L D, 4.8 where C0, k := C 0, Cn, k := 4 n 1 C n k n for n Furthermore, if u n L Ω, H D, there also holds E u n H D 1 k + 1 Cn, ke f c 0 k L D, 4.30 where c 0 := min{1, kc 0 }. 101

118 Proof. For each n 0, the PDE problem associated with u n is the same type Helmholtz problem as the original problem with ε = 0 in the left-hand side of the PDE. Hence, all a priori estimates of Theorem 4..6 hold for each u n with its respective right-hand source side function. First, we have E u 0 L D + u 0 L D + c 0 u 0 L D C 0 k + 1 E f k L D, E u 0 H 1 D C E f k L D. 4.3 Thus, 4.7 and 4.8 hold for n = 0. Without loss of generality we assume that C 0 1. Next, we use induction to prove that 4.7 and 4.8 hold for all n > 0. Assume that 4.7 and 4.8 hold for all 0 n l 1, then E u l L D + u l L D + c 0 u l L D 1 C 0 k + 1 E k ηu k l 1 + δ L D 1l k η u l + kηu L D l 1 L D 1 C 0 k k 4Cl 1, k + Cl, k E f k L D 1 k + 1 8C0 1 + k Cl 1, k 1 + E f k L D 1 k + 1 k Cl, ke f L D, Cl, k Cl 1, k 10

119 where δ 1l = 1 δ 1l and δ 1l denotes the Kronecker delta. To obtain the above result, we note that k, C 0 1 yield the following inequality: 8C k Cl, k Cl 1, k 1 + Cl 1, k = 8C k Cl 1, k 1 + 4l 1 C0 l k l 4 l 1 1 C01 l + k l 1 = 8C k 1 Cl 1, k C k 4 C k Cl 1, k = Cl, k for l. Similarly, E u l H 1 D C E k ηu k l 1 + δ L D 1l k η u l + kηu L D l 1 L D C k 4Cl 1, k + Cl, k E f k L D Cl, ke f k L D. Hence, 4.7 and 4.8 hold for n = l. So the induction argument is complete. 103

120 We now use 4.7 and the elliptic theory for Poisson problems directly to verify estimate E u n H D C 0 k + 1 E k ηu k n 1 + k η u L D n + kηu L D n 1 H 1 D C 0 k + 1 k E 4 u k n 1 L D + u n L D + c 0 u n 1 L kc D 0 C 0 k k 4Cn 1, k + Cn, k E f c 0 k L D 1 k + 1 Cn, ke f c 0 k L D. Hence, 4.30 holds for all n 0. With a priori estimates 4.7 and 4.8 in hand, the proof of existence and uniqueness of each u n follows verbatim the proof of Theorem The proof is complete. Now we are ready to justify the multi-modes representation 4. for the solution u ε of problem To carry this step out we define the partial multi-modes expansion UN ε as N UN ε := ε n u n, n=0 where N is some positive integer. UN ε will also play a key role in our overall numerical approximation method. Since the full series u ε cannot be computed, we approximate u ε by its truncation UN ε. Theorem Let {u n } be the same as in Theorem Then 4. is valid in L Ω, H 1 D provided that σ := 4εC k < 1. Proof. The proof consists of two parts: i the infinite series on the right-hand side of 4. converges in L Ω, H 1 D; ii the limit coincides with the solution u ε. To 104

121 prove i, note for any fixed positive integer p we have U ε N+p U ε N = N+p 1 n=n ε n u n. It follows from the Cauchy-Schwarz inequality and 4.7 that for j = 0, 1 E U ε N+p U ε N H j D Thus, if σ < 1 we have N+p 1 p n=n ε n E u n H j D p k j E f k C 0 p k j E f k L D N+p 1 n=n ε n Cn, k N+p 1 L D σ n n=n C 0 p k j E f k L D σn 1 σ p. 1 σ lim E UN+p ε UN ε H N 1 D = 0. Therefore, {U ε N } is a Cauchy sequence in L Ω, H 1 D. Since L Ω, H 1 D is a Banach space, then there exists a function U ε L Ω, H 1 D such that lim U N ε = U ε N in L Ω, H 1 D. 105

122 To show ii, we first notice that by the definitions of u n and U ε N, au ε N, v = N 1 n=0 N 1 = = = n=0 Ω Ω Ω Ω N 1 n=0 ε n u n, v D k α u n, v D + ik u n, v D dp ε n u n, v D k u n, v D + ik u n, v D dp Ω f, v D dp + N 1 n=0 Thus, U ε N satisfies Ω Ω ε n k εη + ε η u n, v D ik εηu n, v D dp N 1 n=1 Ω ε n k ηu n 1 + k η u n, v D ik ηu n 1, v D dp ε n k εη + ε η u n, v D ik εηu n, v D dp f, v D dp k ε N + ikε N Ω Ω η + εηun 1 + η u N, v D dp ηu N 1, v D dp, U ε N, v k α U ε D N, v + ik αu ε D N, v D dp 4.33 = f, v D dp k ε N η + εηun 1 + η u N, v dp D Ω Ω + ikε N ηu N 1, v D dp Ω for all v L Ω, H 1 D. Where α = 1 + εη. In other words, UN ε Helmholtz problem: UN ε k α UN ε = f k ε N η + εηu N 1 + η u N ν UN ε + ikαun ε = ikε N ηu N 1 solves the following in D, on D. 106

123 By 4.7 and the Cauchy-Schwarz inequality we have k ε N Ω η + εηun 1 + η u N, v D dp 3k ε N E un 1 L D 1 + E u N L D 1 E v L D 1 1 6k ε N k + 1 CN 1, k 1 k E f L D 1 E v L D 1 3εk + 1C 1 0 σ N 1 E f L D 1 E v L D 1 0 as N provided that σ < 1. Similarly, we get kε N Ω ηu N 1, v D dp kε N E u N 1 L D 1 E v L D 1 1 kε N k + 1 CN 1, k E f k L D 1 E v L D 1 ε C 1 k 0 σ N 1 E f L D 1 E v L D 1 0 as N provided that σ < 1. Setting N in 4.33 immediately yields Ω U ε, v k α U ε, v + ik αu ε, v D D D dp = f, v D dp, 4.34 Ω for all v L Ω, H 1 D. Thus, U ε is a solution to problem By the uniqueness of the solution, we conclude that U ε = u ε. Therefore, 4. holds in L Ω, H 1 D. The proof is complete. The above proof also infers an upper bound for the error u ε U ε N next theorem. as stated in the 107

124 Theorem Let U ε N be the same as above and uε denote the solution to problem and σ := 4εC k. Then there holds for εε + 1 < γ 0 E u ε U ε N H j D 9C 0σ N k j + 1 4E f 31 + k L k D, j = 0, 1, 4.35 provided that σ < 1. Where C 0 is a positive constant independent of k and ε. Proof. Let EN ε := uε UN ε, subtracting 4.33 from 4.34 we get Ω E ε N, v k α E ε D N, v + ik D αeε N, v D dp 4.36 = k ε N η + εηun 1 + η u N, v dp D ikεn ηu N 1, v D dp, Ω Ω for all v L Ω, H 1 D. In other words, E ε N solves the following Helmholtz problem: E ε N k α E ε N = k ε N η + εηu N 1 + η u N ν E ε N + ikαe ε N = ikε N ηu N 1 in D, on D. By Theorem 4..6 and 4.7 we obtain for j = 0, 1 E E ε N H j D 18C 0 The proof is complete. k j k [k 4 ε N E u N 1 L D + E u N L D + k ε N E u N 1 L D ] 18C 0 k 4 ε N k j CN 1, k E f k 18C 0σ N k j + 1 4E f k L k D. L D Remark Theorem shows that the error introduced by truncating the multi-modes expansion is on the order of ε N where N is the number of modes in the truncated multi-modes expansion. Since ε is small, U ε N using only a few mode functions, i.e. N is relatively small. 108 can be used to approximate uε

125 4.4 Monte Carlo Discontinuous Galerkin Approximation of the Truncated Multi-modes Expansion U ε N In the previous section, we present a multi-modes representation of the solution u ε and a convergence rate estimate for its truncated multi-modes approximation. These results will serve as the theoretical foundation for our overall numerical methodology for approximating the solution u ε of problem As stated previously, we start by approximating u ε through its truncated multimodes expansion UN ε. Note that the linear nature of the expectation operator, along with the definition of U ε N yields the following expansion: EU ε N = N 1 n=0 ε n Eu n. Hence, to gain an accurate approximation of EUN ε one only needs to seek an accurate approximation of Eu n for each mode function u n. Observe that we can apply the expectation operator to 4.4 and 4.6 to find Eu 0 k Eu 0 = Ef, in Ω, 4.37 ν Eu 0 + ikeu 0 = 0, on Ω Therefore, for Ef known, Eu 0 can be computed directly by solving a deterministic Helmholtz equation. On the other hand, for n 1, we apply the same reasoning to 4.5 and 4.6 to find the following: Eu n k Eu n = k Eηu n 1 + k Eη u n, in Ω, ν Eu n + ikeu n = ikeηu n 1, on Ω. 109

126 We note the terms Eηu n 1 and Eη u n cannot be further broken apart due to the multiplicative nature of these terms and the fact that η and u n are not independent. Thus for n 1, Eu n cannot be computed directly in the same manner as Eu 0. The goal of this section is to develop a Monte Carlo interior penalty discontinuous Galerkin MCIP-DG method for the above mentioned Helmholtz problem. Our MCIP-DG method is the direct generalization of the deterministic IP-DG method proposed in [4, 44] for the related deterministic Helmholtz problem. It should be noted that although various numerical methods such as finite difference, finite element and spectral methods can be used for the job, the IP-DG method presented below is the only general purpose discretization method which is absolutely stable i.e., stable without mesh constraint and optimally convergent. This is indeed the primary reason why we choose this IP-DG method as our spatial discretization method DG Notations To define the IP-DG method used in this chapter, we must introduce some standard DG notation. This notation was first introduced in Chapter 3. Let T h be a quasiuniform partition of D such that D = K T h K. Let h K denote the diameter of K T h and h := max{h K ; K T h }. H s T h denotes the standard broken Sobolev space and V h r denotes the DG finite element space which are defined as H s T h := K T h H s K, V h r := K T h P r K, where P r K is the set of all polynomials whose degrees do not exceed a given positive integer r. Let E I denote the set of all interior faces/edges of T h, E B denote the set of all boundary faces/edges of T h, and E := E I E B. The L -inner product for piecewise functions over the mesh T h is naturally defined by v, w Th := v w dx, K T h K 110

127 and for any set S h E, the L -inner product over S h is defined by v, w Sh := v w ds. e S h e Let K, K T h and e = K K and assume global labeling number of K is bigger than that of K. We choose n e := n K e = n K e as the unit normal on e outward to K and define the following standard jump and average notations across the face/edge e: [v] := v K v K on e E I, [v] := v on e E B, {v} := 1 v K + v K on e E I, {v} := v on e E B for v V h r. We also define the following semi-norms on H s T h : v 1,h,D := v L T h, v 1,h,D := v 1,h,D := v 1,h,D + e E I h + r j=1 e Eh I v 1,h,D + e E I h γ 0,e r h e d 1 [v] L e + he j 1 γ j,e [ j r h e l=1 γ 0,e r { n e v} L e β 1,e r h e n e v] L e [ τ l e v] L e 4.4. IP-DG Method for Deterministic Helmholtz Problem In this subsection, we consider following deterministic Helmholtz problem and its IP-DG approximations proposed in [4, 44]. 1. 1, Φ 0 k Φ 0 = F 0 in D, 4.39 ν Φ 0 + ikφ 0 = G 0 on D

128 Recall that Φ 0 = Eu 0 satisfies the above equations with F 0 = Ef and G 0 = 0. As an interesting byproduct, all the results to be presented in this subsection apply to Eu 0. The IP-DG weak formulation for is defined by cf. [4, 44] seeking Φ 0 H 1 D H r+1 loc D such that a h Φ 0, ψ = F 0, ψ D + G 0, ψ D ψ H 1 D H r+1 T h, 4.41 where a h φ, ψ := b h φ, ψ k φ, ψ Th + ik φ, ψ E B h b h φ, ψ := φ, ψ Th L 1 φ, ψ := e E I h J j φ, ψ := e E I h + i L 1 φ, ψ + { n φ}, [ψ] E Ih + [φ], { n ψ} E Ih, d 1 β 1,e h 1 e [ τ lφ], [ τ lψ] e, l=1 γ j,e h j 1 e [ j n φ], [ nψ] j, j = 0, 1,, r. e r j=0 J j φ, ψ, 4.4 {β 1,e } and {γ j,e } are piecewise constant nonnegative functions defined on E I h. {τ l } d 1 l=1 denotes an orthonormal basis of the edge and τ l the direction of τ l. denotes the tangential derivative in Remark L 1 and {J j } terms are called interior penalty terms, {β 1,e } and {γ j,e } are called penalty parameters. The two distinct features of the DG sesquilinear form a h, are: i it penalizes not only the jumps of the function values but also penalizes the jumps of the tangential derivatives as well the jumps of all normal derivatives up to rth order; ii the penalty parameters are purely imaginary numbers with nonnegative imaginary parts. 11

129 Following [4, 44] and based on the DG weak formulation 4.41, our IP-DG method for problem is defined by seeking Φ h 0 V h r such that a h Φ h 0, ψ h = F 0, ψ h D + G 0, ψ h D ψ h V h r For the above IP-DG method, it was proved in [4, 44] that the method is absolutely stable and its solutions satisfy some wave-number explicit stability estimates. Its solutions also satisfy optimal order in h error estimates, which are described below. Theorem Let Φ h 0 V h r be a solution to scheme 4.43, then there hold i For all h, k > 0, there exists a positive constant Ĉ0 independent of k and h such that Φ h 0 L D + 1 k Φ h 0 1,h,D + Φ h 0 L D Ĉ0C s ˆMF0, G 0, 4.44 where C s := d k + 1 k + 1 r k k max h e + r 5 e Eh I γ 0,e h e + r + r3 β 1,e, h e h e γ 1,e + r γj,e max 4.45 h e 0 j r 1 γ j+1,e ˆMF 0, G 0 := F 0 L D + G 0 L D ii If k 3 h r = O1, then there exists a positive constant Ĉ0 independent of k and h such that Φ h 0 L D + Φ h 0 L D + 1 k Φh 0 1,h,D Ĉ0 1 k + 1 k ˆMF0, G

130 An immediate consequence of 4.44 is the following unconditional solvability and uniqueness result. Corollary There exists a unique solution to scheme 4.43 for all k, h > 0. Theorem Let Φ h 0 V h solve 4.43, Φ 0 H s Ω be the solution of , and µ = min{r + 1, s}. Suppose γ j,e, β 1,e > 0. Let γ j = max e E I γ j,e and λ = γ 0. i For all h, k > 0, there exists a positive constant C 0 independent of k and h such that where Φ 0 Φ h 0 1,h,D C 0 C r + k3 h h µ 1 r C sĉr r Φ 0 s 1 H s D, 4.48 Φ 0 Φ h 0 L D + Φ 0 Φ h 0 L D C h 0 Ĉ r 1 + k µ C s r Φ 0 s H s D, 4.49 C r := λ 1 + r + γ 0 Ĉ r := r j=1 1 + r γ 0 + r γ 1 + r j 1 γ j + kh 1, λr r j= r j γ j + kh 1 C r. λr ii If k 3 h r = O1, then there exists a positive constant C 0 independent of k and h such that Φ0 Φ h 0 1,h,D C 0 r + k hh µ 1 Φ r s 0 H s D, 4.50 Φ0 Φ h 0 Φ0 L Φ h D 0 0 kh µ L Φ D r s 0 H s D Remark It was proved in [7] also by Theorem 4..6 with ε = 0 that Φ 0 H s D C 0 k s k ˆMF0, G 0, s = 0, 1,. 114

131 It is expected that the following higher order norm estimates also hold cf. [4] for an explanation: Φ 0 H s D C 0 k s F 0 H k s D + G 0 H s 5, s D provided that F 0, G 0 and D are sufficiently smooth. In such a case, Φ 0 H s D in can be replaced by the above bound so explicit constants can be obtained in these estimates. Theorems 4.4. and give stability and optimal error estimates in h for the IP-DG method in both the pre-asymptotic and asymptotic regime. Here the asymptotic regime is characterized as k, h, r chosen to satisfy the constraint k 3 h r = O1. In the remainder of this chapter we only consider the asymptotic regime of k 3 h r = O1. This choice was made because the constants for the asymptotic regime in both Theorem 4.4. and are more tractable MCIP-DG Method for Approximating EU ε n We recall that each mode function u n satisfies the following Helmholtz problem: u n k u n = S n in D, 4.53 ν u n + iku n = Q n on D, 4.54 where u 1 := 0 and S 0 :=f, Q 0 := 0, 4.55 S n :=k ηu n 1 + k η u n, Q n := ikηu n 1, 4.56 for n 1. Clearly, S n, x and Q n, x are random variables for a.e. x D, S n L Ω, L D and Q n L Ω, L D. We remark again that due to its multiplicative structure ES n and EQ n can not be computed directly for n

132 Otherwise, 4.53 and 4.54 would be easily converted into deterministic equations for Eu n, as we did early for Eu 0. In other words, is a genuine random PDE problem. On the other hand, since all the coefficients of the equations are constants, then the problem is nearly deterministic. Such a remarkable property will be fully exploited in our overall numerical methodology which will be described in the next section. Several numerical methodologies are well known in the literature for discretizing random PDEs, Monte Carlo Galerkin and stochastic Galerkin or polynomial chaos methods and stochastic collocation methods are three of well-known methods cf. [1, 11] and the references therein. Due to the nearly deterministic structure of , we propose to discretize it using the Monte Carlo IP-DG approach which combines the classical Monte Carlo method for stochastic variable and the IP- DG method, which is presented in the proceeding subsection, for the spatial variable. Following the standard formulation of the Monte Carlo method cf. [1], let M be a large positive integer which will be used to denote the number of realizations and V h r be the DG space defined in Section For each j = 1,,, M, we sample i.i.d. realizations of the source term fω j, and random medium coefficient ηω j,, and recursively find corresponding approximation u h nω j, V h r such that a h u h n ω j,, ψ h = S h nω j,, ψ h D + Qh nω j,, ψ h D ψ h V h r 4.57 for n = 0, 1,,, N 1. Where S h 0 ω j, := fω j,, Q h 0 := 0, 4.58 u h 1ω j, := 0, 4.59 S h nω j, := k ηu h n 1ω j, + k η u h n ω j,, n = 1,,, N 1, 4.60 Q h nω j, := ikηu h n 1ω j,, n = 1,,, N

133 We point out that in order for u h n to be computable, Sn h and Q h n, not S n and Q n, are used on the right-hand side of This small perturbation on the right-hand side will result in an additional discretization error which must be accounted for later. We approximate Eu n by the following sample average Φ h n := 1 M M u h nω j,. 4.6 j=1 Thus, EUN ε can be approximated by Ψ h N = N 1 n=0 ε n Φ h n The rest of this section is used to analyze the error generated by this MCIP-DG method. This will be carried out in the following two steps: i We estimate the error from the IP-DG approximation, i.e. U ε N U h N, where U h N = N 1 n=0 εn u h n. ii We estimate the error from the Monte Carlo method, i.e. EU h N Ψh N. We begin by obtaining stability estimates for each u h n. Lemma Assume k 3 h r = O1. Then for each n 0 the following stability estimate holds: E u h n L D + uh n L D k E u h n 1,h,D 1 k + 1 k Ĉn, k E f L D, 4.64 Ĉn, k E f L D, 4.65 where Ĉ0, k := Ĉ 0, Ĉn, k := 4 n 1 Ĉ n k n for n

134 Proof. Using estimate 4.47 one immediately obtains E u h 0 L D + uh 0 L D Ĉ 0 E u h 0 1,h,D Ĉ 0 1 k + 1 k E S h 0 L D Ĉ k E S h 0 L D Ĉ 0 1 k + 1 k E f k L D, E f L D, which verifies 4.64 and 4.65 for n = 0. Suppose 4.64 and 4.65 hold for all n = 0, 1,,, l 1. It remains to show 4.64 and 4.65 hold for n = l. Using 4.64 with n = l 1 and steps that were used previously in the proof of Theorem one obtains the following: E u h l L D + uh l L D 1 Ĉ 0 k + 1 E S h k l L D + Qh n L D 1 0 Ĉ k + 1 k 4 E 4 u h k l 1 L D + uh l L D + 1 k uh l 1 L D 1 Ĉ 0 k k 4Ĉl k 1, k + Ĉl, k E f L D 1 0 8Ĉ k k Ĉl, k Ĉl 1, k 1 + E f k L 4Ĉl 1, k D 1 k + 1 Ĉl, ke f k L D. Here we have used the fact that 8Ĉ 01 + k Ĉl, k Ĉl 1, k 1 + Ĉl, k. 4Ĉl 1, k Thus 4.64 is proved for n = l. 118

135 Using 4.65 along with a similar argument the following is found: E u h l 1,h,D Ĉ E Sl h L k D + Qh n L D Ĉ k 4 E 4 u h k l 1 L D + E uh l L D + 1 k uh l 1 L D Ĉ k 4Ĉl k 1, k + Ĉl, k E f L D Ĉl, ke f L k D. Hence 4.65 holds for n = l. Therefore, the induction argument is complete. To complete step i, we need a set of auxiliary mode functions { ũ h n fixed realizations ηω j, and fω j,, we define ũ h nω j, V h r following problem: } n 0. For as the solution to the a h ũωj, h n, ψ h = S n ω j,, ψ h D + Q nω j,, ψ h D ψ h V h r S n ω j, and Q n ω j, were defined in 4.55 and 4.56 and are different from Sn h and Q h n that are used in the definition of u h n. We also need the following lemma. Lemma Let γ, β > 0 be two real numbers, {c n } n 0 and {α n } n 0 be two sequences of nonnegative numbers such that c 0 γα 0, c n βc n 1 + γα n for n Then there holds n c n γ β n j α j for n j=0 The proof to this lemma is trivial and thus is omitted. Now, we are ready to estimate the error u n u h n. 119

136 Lemma Suppose k 3 h r = O1. Then the following error estimates hold: E u n u h n L D + u n u h n L D n C 0 kh µ r s j=0 E u n u h n 1,h,D C C 0k1 + kh µ 1 where µ = min{r + 1, s}. r s [ C0 k + 3 ] n j E uj H s D, 4.70 n [ C0 k + 3 ] n j E uj H s D, 4.71 Proof. To begin, we introduce the following error decomposition: j=0 u n u h n = u n ũ h n + ũh n u h n. Thus, we get estimates on the error u n u h n by first estimating u n ũ h n and then estimating ũ h n u h n. As an immediate consequence of Theorem part ii, for k 3 h r = O1 the following estimates hold: E un ũ h C0 r + k hh µ 1 n 1,h,D E u n H s D, 4.7 r s E u n ũ h n + L u D n ũ h n L D C0 kh µ E u n H s D To bound ũ h n u h n, we observe that subtracting 4.57 from 4.67 yields r s a h ũh n u h n, ψ h = S n S h n, ψ h D + Q n Q h n, ψ h D ψ h V h r, a.s. 10

137 We apply Theorem 4.4. ii to obtain the following estimate: E k ũ hn u hn L D + k ũ hn u hn L D + ũ hn u hn 1,h,D Ĉ E S n S h k n L D + Q n Q h n L D C 0 kk + 1E u n 1 u h n 1 L D + u n u h n L D + u n 1 u h n 1 L D Combining 4.73 and 4.74 and applying the triangle inequality, we get E u n u h n L D + u n u h n L D E ũ h n u h n L D + ũ h n u h n L D + u n ũ h n L D + u n ũ h n L D C 0 k + 1E u n 1 u h n 1 L D + u n u h n L D u n 1 u h n 1 L D + C 0 kh µ r s E u n H s D. To estimate the error in 1,h,D, we apply an inverse inequality along with 4.73, 4.74, and the triangle inequality to get E u n u h n 1,h,D E ũ h n u h n 1,h,D + u n ũ h n 1,h,D Ch 1 E ũ h n u h n L D + E un ũ h n 1,h,D C C 0 h 1 k + 1E u n 1 u h n 1 L D + u n u h n L D k u n 1 u h n 1 L D + C 0 r + k hh µ 1 r s E u n H s D. 11

138 Thus, 4.75 and 4.76 give recursive estimates for the error u n u h n. Next, we note the following estimates: and define E u 1 u h 1 L D = E u 1 u h 1 1,h,D = 0, 4.77 E u 0 u h 0 L D + u 0 u h C0 kh µ 0 L D E u 0 H s D, 4.78 E u 0 u h C0 r + k hh µ 1 0 1,h,D E u 0 H s D, 4.79 r s r s u = u 1 = u h = u h 1 = 0, c n : = E u n u h n L D + u n 1 u h n 1 L D + E u n u h n L D + u n 1 u h n 1 L D, β : = C 0 k + 3, γ := C 0 kh µ, α r s n := E u n H s D. Then by 4.76 these defined quantities meet the assumptions in Lemma Applying Lemma yields produce Now 4.76 and 4.70 can be combined to Now Lemma can be used to bound the error due to IP-DG discretization, i.e. U ε N U h N. Theorem Assume that u n L Ω, H s D for n 0. Then the spatial error U ε N U h N satisfies the following estimates: E UN ε UN h C0 kh µ N 1 L D r s n=0 E UN ε UN h 1,h,D C C 0k1 + kh µ 1 r s N 1 n ε n[ C0 k + 3 ] n j E uj H D s j=0 n=0 j=0 n ε n[ C0 k + 3 ] n j E uj H D. s

139 To simplify the above spatial error estimates, bounds for E u n H s D in terms of higher order norms of f are necessary. Only the case s = is considered below for simplicity. When s =, the required estimates have been obtained in These estimates in conjunction with Theorem yield the following results: Theorem Assume that u n L Ω, H D for n 0. Then the following estimates hold: E U ε N U h N L D C3 N, k, ε h f L Ω,L D, 4.8 E U ε N U h N 1,h,D C4 N, k, ε h f L Ω,L D, 4.83 where C 3 N, k, ε := C 0 k r C 4 N, k, ε := C C 0k1 + k r C 0 k k C C0 C0 k + 3ε N 1 C 0 C0 k + 3ε, 4.84 C 0 k k C C0 C0 k + 3ε N 1 C 0 C0 k + 3ε Proof. To obtain the above result, we need to estimate the double sum N 1 n=0 j=0 n ε n[ C0 k + 3 ] n j E uj H D, 13

140 which is in Theorem Applying 4.30 to estimate u j H D and exploiting some geometric series properties of the sum we get N 1 n=0 j=0 n ε n[ C0 k + 3 ] n j E uj H s D k + 1 N 1 f k L Ω,L D = C 1 0 k k C 1 0 k k f L Ω,L D f L Ω,L D n n=0 j=0 N 1 n=0 j=0 N 1 n=0 C 0k k C 0 1 f L Ω,L D ε n[ C0 k + 3 ] n j Cj, k 1 n ε n 4 j C j k j[ C0 k + 3 ] n j ε n[ C0 k + 3 ] n N 1 n=0 n j=0 j C j 0 [ ε C0 C0 k + 3 ] n C 0k k C C0 C0 k + 3ε N 1 C 0 C0 k + 3ε f L Ω,L D. We get 4.8 and 4.83 by applying the above inequality to 4.80 and 4.81 respectively. Remark Theorem , shows that the error generated by the proposed IP-DG method is optimal in the mesh size h. With i complete, we turn our attention to ii, i.e. estimating the error generated by using the Monte Carlo method. The following lemma is well known cf. [1, 58]. Lemma For n 0 the following estimates hold: E Eu h n Φ h n 1 L D M E uh n L D, 4.86 E Eu h n Φ h n 1 1,h,D M E uh n 1,h,D Combining Lemmas and 4.4.6, we get the following error estimate theorem. 14

141 Theorem Under the constraint k 3 h r = O1 the following estimates hold: E Eu h n Φ h n 1 L D M E Eu h n Φ h n 1 1,h,D M 1 k + 1 k Ĉn, k E f L D, Ĉn, k E f L k D, 4.89 Remark Estimates 4.88 and 4.89 show that for each fixed n 0 the statistical error due to sampling is controlled by the number of realizations of u h n. Indeed, it can be easily proved by using Markov s inequality and Borel-Cantelli lemma that the statistical error converges to zero as M tends to infinity, see [1, Proposition 4.1] and [58, Theorem 3.]. The above estimates on Eu h N Φh n are now used to obtain the following theorem. Theorem Suppose k 3 h r = O1 and ˆσ := 4εĈ01 + k < 1. Then the following estimates hold: E EU h N Ψ h N L D Ĉ 0 M E EU h N Ψ h N 1,h,D Ĉ 0 M 1 k + 1 k f L Ω,L D k f L Ω,L D 1 1 ˆσ, ˆσ Proof. First, note that U h N Ψ h N = N 1 n=0 ε n u h n Φ h n. 15

142 By 4.88 we get E EU h N Ψ h N L D N 1 n=0 ε n E Eu h n Φ h n L D 1 1 M k + 1 N 1 f k L Ω,L D ε n Ĉn, k 1 where ˆσ := 4εĈ01 + k < 1. Similarly, by 4.89 Ĉ0 M Ĉ0 M n=0 1 k + 1 N 1 f k L Ω,L D 4 n ε n Ĉ0 n 1 + k n 1 k + 1 k f L Ω,L D n=0 1 1 ˆσ, E N 1 EU h N Ψ h N 1,h,D ε n E Eu h n Φ h n 1,h,D The proof is complete. n=0 Ĉ0 M f L k Ω,L D 1 1 ˆσ. Remark Theorem shows that the error generated by using the Monte Carlo method is on the order of OM 1. Thus, for an accurate approximation a large number of realizations M must be taken. 4.5 The Overall Numerical Procedure This section is devoted to defining and analyzing the overall efficient MCIP-DG algorithm for computing u ε. The key to efficiency is the exploitation of the special structure inherent in the multi-modes expansion. In Subsection 4.5.1, the efficient MCIP-DG algorithm is defined and its computational complexity is analyzed. Subsection 4.5. summarizes all of the error estimates given in the previous sections 16

143 to obtain an estimate of the total error produced by using the multi-modes MCIP-DG method The Numerical Algorithm, Linear Solver and Computational Complexity The goal of this subsection is to introduce an efficient MCIP-DG method to approximate the expectation of the solution to the random Helmholtz problem The key to this method is the exploitation of the special structure of the multi-modes expansion of the solution described in Section 4.3. In order to judge the efficiency of this method, we must establish a reliable standard upon which to compare and contrast our method. For this standard, we use the classical MCIP-DG method that does not utilize the multi-modes expansion of the solution. In order to define such a method, we need to introduce a new IP-DG formulation. Given a sample realization of the coefficient ηω j, and source data fω j, define û h ω j, V r h the solution to as â h j û h ω j,, v h = fω j,, v h D, v h V r h, 4.9 where â h j φ, ψ := b h φ, ψ k 1 + εηω j, φ, ψ + ik 1 + εηω T j, φ, ψ h E B h r + i L 1 φ, ψ + J m φ, ψ. m=0 Here b h,, L 1,, and J m, are defined previously in Subsection Notice that the main difference between 4.9 and 4.57 which was used to define u h n is that the sesquilinear form â h j, depends on the realization ηω j,. This is the key observation that makes the use of the multi-modes expansion worth-while when seeking an efficient MCIP-DG method. 17

144 Based on 4.9 the classical MCIP-DG method for solving the random Helmholtz problem is defined by the following algorithm: Algorithm 1 Classical MCIP-DG Inputs: f, η, ε, k, h, M. Set Ψ h = 0 initializing. For j = 1,,, M Obtain realizations ηω j, and fω j,. Solve for û h ω j, V h r such that â h j ûh ω j,, v h = fωj,, v h D v h V h r. Set Ψ h Ψ h + 1 M ûh ω j,. Endfor Output Ψ h. This algorithm is very expensive for M large, because at each step of the loop a deterministic Helmholtz equation must be solved. This requires one to solve a large especially for k large, ill-conditioned, indefinite linear system. It is well-known that no standard iterative method works well for such a system [37]. For this reason, Gaussian elimination is considered for each solve in the loop. Since the Gaussian elimination step is the most costly portion of the loop, the computational complexity is estimated in terms of Gaussian elimination steps. Let h be the mesh size of a quasi-uniform partition T h of the domain D. Then each coefficient matrix that appeared in the for-loop of Algorithm 1 has approximate size OL d L d, where L = 1. Each Gaussian elimination solve will have computational h 3L complexity O 3d. Thus, the overall computational complexity of Algorithm 1 is O. Recall that the Monte Carlo method converges at a rate of OM 1. 3L 3d M 18

145 Thus, M must be chosen sufficiently large in order to gain sufficient error reduction. Therefore, a computational complexity of O is quite costly, and this makes Algorithm 1 not practical. 3L 3d M For this reason, the following algorithm is introduced. Algorithm Multi-Modes MCIP-DG Inputs: f, η, ε, k, h, M, N Set Ψ h N = 0 initializing. Generate the coefficient matrix A associated to the sesquilinear form a h, over V h r V h r. Compute and store the LU decomposition of A. For j = 1,,, M Obtain realizations ηω j, and fω j,. Set S h 0 ω j, = fω j,. Set Q h 0ω j, = 0. Set u h 1ω j, = 0. Set U h N ω j, = 0 initializing. For n = 0, 1,, N 1 Solve for u h nω j, V h r such that a h u h n ω j,, v h = S h n ω j,, v h D + Q h nω j,, v h D v h V h r, using the LU decomposition and forward and backward substitution. Set UN h ω j, UN h ω j, + ε n u h nω j,. Set Sn+1ω h j, = k ηω j, u h nω j, + k ηω j, u h n 1ω j,. Set Q h n+1ω j, = ikηω j, u h nω j,. 19

146 Endfor Set Ψ h N Ψh N + 1 M U h N ω j,. Endfor Output Ψ h N. The key difference between Algorithm 1 and Algorithm is the fact that the bilinear form a h, used in the nearly deterministic Helmholtz equation in the inner for loop of Algorithm does not depend on the current mode number n or the current realization number j. Thus, only one stiffness matrix A must be computed and its LU decomposition can be reused when seeking a solution to the equation in the inner loop. This results in a great savings in terms of computational time required by the algorithm. To analyze Algorithm, again the coefficient matrix A will have approximate size OL d L d. Thus, Gaussian elimination used to produce the LU decomposition has order O 3L3d. After this LU decomposition is computed, solving the system using forward and backward substitution has complexity order OL d. Thus, the 3L computational complexity for Algorithm is on the order O 3d + MNL. d M will be chosen large c.f. Remark On the other hand, N will be chosen to be a small positive integer c.f. Remark With the intent of choosing M large, using M = L d 3L for Algorithm yields a computational complexity O 3d + NL. 3d This is on the same order as a few Gaussian elimination solves. The Monte Carlo method is naturally parallelizable and it is in this setting that Algorithm 1 should be implemented. The structure of Algorithm also allows parallel implementation. This being said, unless one uses computational resources in which all Gaussian elimination solves in Algorithm 1 can be carried out at the same time, Algorithm should be more efficient in terms of computation time. 130

147 4.5. Convergence Analysis The goal of this subsection is to analyze the error of the multi-modes MCIP-DG approximation produced by Algorithm. Recall that Algorithm uses the following three sequential approximations: Approximation of u ε with a partial multi-modes expansion U ε N Approximation of U ε N with its IP-DG approximation U h N Approximation of EU h N with its Monte Carlo approximation Ψh N Thus, the error Eu ε Ψ h N following manner: associated with Algorithm can be decomposed in the Eu ε Ψ h N = Eu ε EU ε N + EU ε N EU h N + EU h N Ψ h N. Each piece of this error decomposition has already been estimated in the previous sections of this chapter. The following theorem puts these results together to obtain estimates for the total error of Algorithm : Theorem Under the assumptions that u n L Ω, H D for n 0, k 3 h r = O1 and σ, ˆσ < 1 i.e. ε = Ok 1, the following error estimates hold: E Eu ε Ψ h N L D C 1 ε N + C h + C 3 M 1, 4.93 E Eu ε Ψ h N H 1 D C 4 ε N + C 5 h + C 6 M 1, 4.94 where C j = C j C 0, Ĉ0, k, ε are positive constants for j = 1,,, 6. Proof. To begin, we apply the triangle inequality to the error decomposition given above. Then each term can be estimated separately using Theorems 4.3.3, 4.4.9, and Note that Theorem cannot be used directly; instead, the Cauchy-Schwarz 131

148 inequality must be used in the following manner: E u ε UN ε H D j E 1 E 9C 0σ N 31 + k u ε UN ε H j D k j E f L k D. Thus, taking the square root on both sides and applying the definition of σ yields the first term in 4.93 and With this result and those listed above, the desired inequalities follow. 4.6 Numerical Experiments In this section we present a series of numerical experiments in order to accomplish the following: compare our MCIP-DG method using the multi-modes expansion to a classical MCIP-DG method illustrate examples using our MCIP-DG method in which the perturbation parameter ε satisfies the constraint required by the convergence theory illustrate examples using our MCIP-DG method in which the perturbation parameter constraint is violated In all our numerical experiments we use the spatial domain D = 0.5, 0.5. To partition D we use a uniform triangulation T h. For a positive integer n, T 1/n denotes the triangulation of D consisting of n congruent isosceles triangles with side lengths 1/n, 1/n, and /n. Figure 4.1 gives the sample triangulation T 1/10. To implement the random noise η, we note that η only appears in the integration component of our computations. Therefore, we made the choice to implement η only at quadrature points of the triangulation. To simulate the random media, we let η, ˆx be an independent random number chosen from a uniform distribution on 13

149 Figure 4.1: Triangulation T 1/10. some closed interval at each quadrature point ˆx. Figure 4. shows an example of such random media. Figure 4.: Discrete average media 1 M M j=1 αω j, left and a sample media αω, right computed for h = 1/0, ε = 0.1, η, x U[ 1, 1], and M = MCIP-DG with Multi-modes Expansion Compared to Classical MCIP-DG The goal of this subsection is to verify the accuracy and efficiency of the proposed multi-modes MCIP-DG method. As a benchmark we compare this method to the classical MCIP-DG i.e. produced using Algorithm 1. Throughout this section Ψ h is used to denote the computed approximation to Eu using the classical MCIP-DG. In this subsection, we set f = 1, k = 5, 1/h = 50, M = 1000, and ε = 1/k + 1. Here ε is chosen with the intent of satisfying the constraint set by the convergence 133

150 theory in the preceding section. η is sampled as described above from a uniform distribution on the interval [0, 1]. Ψ h N is computed for N = 1,, 3, 4, 5. In our first test, we compute Ψ h N Ψ h L D. The results are displayed in Figure 4.3. As expected, we find that the difference between Ψ h N and Ψ h is very small. We also observe that we are benefited more by the first couple modes while the help from the later modes is relatively small. From this experiment, we see the error decrease is similar to ε N. This is expected from Theorem To test the efficiency of our MCIP-DG method with multi-modes expansion, we compare the CPU time for computing Ψ h N and Ψ h. Both methods are implemented on the same computer using Matlab. Matlab s built-in LU factorization is called to solve the linear systems. The results of this test are shown in Table 4.1. As expected, we find that the use of the multi-modes expansion improves the CPU time for the computation considerably. In fact, the table shows that this improvement is an order of magnitude. Also, as expected, as the number of modes used is increased the CPU time increases in a linear fashion. Figure 4.3: left Relative error in the L -norm between Ψ h N computed using the multimodes MCIP-DG method and Ψ h computed using the classical MCIP-DG method. right ε N vs. N for N = 1,,, More Numerical Tests The goal of this subsection is to demonstrate the approximations obtained by our multi-modes MCIP-DG method using different magnitudes of parameter ε. We only consider the case 0 < ε < 1 in order to legitimize the series expansion u ε. Our 134

151 Table 4.1: CPU times required to compute the multi-modes MCIP-DG approximation Ψ h N and the classical MCIP-DG approximation Ψ h. Approximation CPU Time s Ψ h Ψ h Ψ h Ψ h Ψ h Ψ h hope is that ε = Ok 1 required by the convergence theory c.f. Theorem is not sharp in practice, and thus our multi-modes MCIP-DG method produces good approximations for larger values of ε. Similar to the numerical experiments from [4], we choose the function f = sin kαω, r /r, where r is the radial distance from the origin and αω, is implemented as described in the beginning of this section. Since our intention is to observe what happens as we vary ε, we fix k = 50, h = 1/100, and M = In Figures 4.4 and 4.5, we set ε = 0.0 and η 1. In Figure 4.4 we present plots of the magnitude of the computed mean Re Ψ3 h and a computed sample Re U h 3, respectively, over the whole domain D. Figure 4.5 gives the plots of a cross section of the computed mean Re Ψ3 h and a computed sample Re U h 3, respectively, over the line y = x. In this first example, we observe that the computed sample does not differ greatly from the computed mean because ε is very small. In Figures , we fix η 1 and increase ε past the constraint established in the preceding convergence theory. As expected, we see that as ε increases the computed sample differs more from the computed mean. We also observe that as ε increases the phase of the wave remains relatively intact but the magnitude of the wave becomes more uniform. In Table 4., the relative error measured in the L -norm between the multimodes approximation Ψ h N and the classical Monte Carlo approximation Ψh is given for ε = 0.0, 0.1, 0.5, 0.8. In this table only three modes i.e., N = 3 are used. Recall 135

152 that the convergence theory in this case only holds for ε on the order of the first value 0.0. That being said, we observe that the approximations corresponding to ε = 0.1 and ε = 0.5 are relatively close to those obtained using the classical Monte Carlo method. Another observation that can be made from Table 4. is that as ε increases the relative error increases. This is expected from the convergence theory. Recall that the error predicted in the convergence theory can be bounded by a term with the factor ε N. Thus for ε relatively large, one must use more modes to decrease the error. Keeping this in mind, Table 4.3 records the relative error measured in the L -norm between the multi-modes approximation Ψ h N and the classical Monte Carlo approximation Ψ h for ε = 0.5, 0.8 and N = 4, 5, 6, 7. We observe that the relative error decreases as N increases when ε = 0.5. On the other hand, the relative error increases as N increases when ε = 0.8. From Tables 4. and 4.3, we observe that multimodes approximation Ψ h N is relatively accurate measured against an approximation from the classical Monte Carlo method even in cases when ε does not satisfy the constraint set forth in the convergence theory. We also observe that when ε becomes too large, the multi-modes approximation no longer agrees with the classical Monte Carlo method. Table 4.: Relative error in the L -norm between the multi-modes MCIP-DG approximation Ψ h 3 and the classical MCIP-DG approximation Ψ h. ε Relative L Error Table 4.3: Relative error in the L -norm between the multi-modes MCIP-DG approximation Ψ h N and the classical MCIP-DG approximation Ψ h. ε N = 4 N = 5 N = 6 N =

153 Figure 4.4: Re Ψ h 3 left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.0, η, x U[ 1, 1], and M = Figure 4.5: Cross sections of Re Ψ h 3 left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.0, η, x U[ 1, 1], and M = 1000, over the line y = x. Figure 4.6: Re Ψ h 3 left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.1, η, x U[ 1, 1], and M =

154 Figure 4.7: Cross sections of Re Ψ h 3 left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.1, η, x U[ 1, 1], and M = Figure 4.8: Re Ψ h 3 left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.5, η, x U[ 1, 1], and M = Figure 4.9: Cross sections of Re Ψ h 3 left and Re U h 3 right computed for k = 50, h = 1/100, ε = 0.5, η, x U[ 1, 1], and M =