Unicorns: Application of Harmonic Coordinates to the Acoustic Wave Equation on Regular Grids in 2D

Transcription

1 RICE UNIVERSITY Unicorns: Application of Harmonic Coordinates to the Acoustic Wave Equation on Regular Grids in 2D by Tommy L. Binford, Jr. A Thesis Proposal Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved, Thesis Committee: William W. Symes, Chairman Noah G. Harding Professor of Computational and Applied Mathematics P. R. Ofessor Assistant Professor of Computational and Applied Mathematics Colin A. Zelt Professor of Earth Science Houston, Texas December 2009

2 Contents List of Figures List of Tables iv v 1 Introduction Organization of the Proposal HCE-FEM for 1D Elliptic Interface Problems Model Problem: Theory Standard FEM Approximation Spatial Discretization HCE Finite Element Method ρ-harmonic Coordinates HCE Finite Element Basis Equivalence of IFEM and HCE-FEM in 1D Stiffness Matrix Mass Matrix Numerical Examples HCE-FEM Applied to 1D Acoustic Wave Equation Model Problem Discretization Spatial Approximation Time Approximation Numerical Examples HCE Finite Elements for 2D Elliptic Problems Model Problem Preliminaries Notational and Semantic Conventions Q 1 Lagrange Finite Elements Reference Map Harmonic Grid ii

3 iii 4.4 2D HCE-FEM Composed Basis Differentiating the HCE-FEM Basis Functions Integrating the HCE-FEM Basis Functions HCE-FEM Algorithm Numerical Example Proposal 41 Bibliography 42

4 List of Figures 2.1 Cartoon of the basis functions in the harmonic coordinate grid Cartoon of the basis functions mapped from the harmonic coordinate grid back to the regular grid. The kinks occur where the flux jump is forced to zero when the density is discontinuous. This introduces regularity into the solution The Q 1 Lagrange basis functions. Note that the each function is unity at a single node The bilinear reference map T K : K K. Note the order of the nodes is preserved The reference map T F (K). A depiction of a true harmonic grid element is shown in blue with the quadrilateral approximation given by the convex hull of the nodes An example of the support for the composite basis functions on a single element. Element K is shown with vertices a i with the basis support K represented by a fine triangular grid. Here the denisity ratio is 1/ An example of the support for the composite basis functions on a single element. Element K is shown with vertices a i with the basis support K represented by a fine triangular grid. Here the denisity ratio is

5 List of Tables 2.1 L 2 error and average rates for the stiffness matrix test problem L 2 error and average rates for the mass matrix test problem with σ L /σ R = L 2 error and average rates for the mass matrix test problem with σ L /σ R = 1/ Error rate for acoustic wave equation test with data from Symes and Vdovina (2008) with and without mass lumping. The L 2 -error is measured in the spatial grid and only involves the reflected wave Relative RMS error (%) for acoustic wave equation test with wave speeds from Symes and Vdovina (2008), high density contrast, and with and without mass lumping Error rate for elliptic test problem. The error is measured in percentage relative to the exact solution in the l 2 -norm

6 Chapter 1 Introduction The finite element method (FEM) has been used extensively to solve a wide variety of problems in computational physics. Models of natural structures often have irregular and discontinuous material properties that vary over a continuum of scales. A grid, or mesh, that conforms to the material variation is impractical for realistic problems due to the computational cost. Removing the option for conforming grids, special consideration is required in the discrete setting due to limitations of the underlying approximation spaces. Material transitions and fluxuations within elements reduce the regularity of the solution. A typical problem exhibiting such characteristics is the interface problem encountered in seismic modeling. Using regular grids, material transitions occur within elements or grid cells resulting in an error due to misalignment. There are two main methods used to solve computational physics problems, and in particular, wave propagation models. Both require modification to accomodate the grid misalignment problem. A brief description of each follows. The finite difference method is the current state-of-the-art and industry standard. The inability of FDM to accurately resolve interfaces that do not conform to the computational grid has been quantified and the error explicitly separated into that governed by truncation and the more sinister and obstinate misalignment error (Brown, 1984). Further examined in the context of high order finite difference time domain method, the first order error component is arguably suppressed in the preasymptotic regime; however, the error can only be ignored under certain conditions rather than eliminated (Gustafsson and Wahlund, 2004). More recently, the truly catastrophic repercussions of grid misalignment error were demonstrated (Symes and Vdovina, 2008). Without modifying the approximation space, there can be no hope of recovering the solution. Such is the aim of the immersed interface method (IIM) and related methods (Leveque and Li, 1994; Zhang and Leveque, 1997; Zhou et al., 2006). It is obvious from the construction that subgrid inclusions and highly variable coefficients were never intended by these techniques. Interest in applying the finite element method to elliptic problems with discontinu- 1

7 ous coefficients has continued for more than four decades (Babuska, 1970). A detailed treatment of the problem with a variety of historical notes can be found in Maugeri et al. (2000). The seriousness of the misaligned interface problem is evident by simply noting the wealth of extensions to FEM, such as the early nullspace scaled hybrid isotropic (NSHI) method (Vavasis, 1996), immersed FEM (IFEM) (Li, 1998; Li et al., 2003; Li and Ito, 2006; Kafafy et al., 2005), extended FEM (XFEM) (Möes et al., 1999), and the more recent conformal decomposition FEM (Noble et al., 2009). The nature of the approximation issue is equivalent to that noted by the FDM researchers. Interestingly, none of the FEM methods mentioned successfully treats the problem of highly varying coefficients. However, it should be noted that each numerical scheme has strong theoretical and numerical results for their respective special cases. Yet each method seeks to enrich the approximation space in a similar manner. The foundation of FEM in variational calculus makes it arguably superior to FDM as far as theoretical work is concerned. Thus, we choose to focus on a modification to FEM. Recently, theoretical work using harmonic maps has lead to a finite element method based on simplicial elements that recovers the optimal rate of convergence without ad-hoc basis enrichment (Owhadi and Zhang, 2006; Zhang, 2007). The genesis of the coordinate transformation idea lay in early work in the field of homogenization (Bensoussan et al., 1978). The notion of a coordinate transformation arises naturally and has even been used and noted, albeit without justification, independent of homogenization (Bamberger et al., 1979). Alessandrini and Nesi (2001) provided the impetus for Owhadi s reseach, where upscaling in the spirit of Bamberger, et al. is no longer restricted to periodic materials. Further work by Alessandrini and Nesi (2003) has revealed a connection between the harmonic coordinate transformation and quasiconformal maps. A feature of this new method is coefficient variations are not limited to interfaces described by conventional notion of an interface. In fact, the only restriction is the coefficients must be bounded and measurable. The promise of this method is great, but all numerical results to date fail to exhibit the expected rates of convergence (Owhadi and Zhang, 2006; Zhang, 2007). The limitation is caused by inaccurate representation of the basis function support. The key issue remaining to make this method practically useful is accurate representation of the distorted elements and accurate integration of the basis functions over these elements. The aim of this work is to investigate and resolve the integration issue related to the application of the harmonic coordinate enhanced finite element method (HCE- FEM). In keeping with the existing finite difference framework developed by The Rice Inversion Project, we apply the method to quadrilateral elements to establish a 2D finite element method using regular grids as a replacement for the finite difference spatial discretization. Restriction to regular grids eliminates the challenge of grid generation as this work will eventually extend to 3D where automatic grid generation is an active area of research. Once implemented, our method will provide numerical results in line with theoretical estimates, meanwhile opening another area 2

8 3 of investigation into the accuracy needed for the harmonic map problem to deal with efficiency. 1.1 Organization of the Proposal In the next chapter, the finite element method with harmonic coordinates is first applied to an elliptic equation in 1D with a single interface. The results shown highlight the ability of this technique to preserve optimal order accuracy in spite of large contrasts in material properties, particularly density. An application to the acoustic wave equation in 1D follows. This demonstration shows that mass lumped FEM and HCE-FEM may be used interchangeably for constant density acoustics. In Chapter 4, the method is applied to an elliptic equation in 2D with piecewise constant coefficients. Numerical results revealing the convergence issues are shown to emphasize the need for an accurate integration method. In the last chapter, we propose to solve 2D scalar acoustic wave equation on regular grid using the algorithm outlined in Chapter 4.

9 Chapter 2 HCE-FEM for 1D Elliptic Interface Problems In this chapter, we present apply the finite element method to an elliptic test problem with piecewise constant coefficients. We are particularly interested in the acoustic wave equation, and the elliptic test problem leads naturally to the spatial discretization obtained by applying the finite element method to that problem. The finite element function space is replaced by a piecewise linear basis composed with a harmonic coordinate map. The algorithm presented in this chapter provides both a a starting point and justification for 2D application. After defining the harmonic coordinate enhanced basis, it is shown that the resulting composite basis functions are exactly those of the immersed finite element method. However, this equivalence is only true in 1D. The modified mass and stiffness matrices are then assembled based on the standard discretization. Examples comparing FEM and the new method are provided. 2.1 Model Problem: Theory Let Ω R. Consider the general scalar elliptic PDE in 1D ( ) 1 u + η u + βu = f in Ω, (2.1) x ρ x x u = 0 on Ω, (2.2) where the coefficients ρ, η, β L (Ω), f L 2 (Ω), and the solution u : Ω R. With ρ > 0, the differential operator in Eq. (2.1) satisfies the definition of ellipticity. To simplify the model, let η = 0. This is justified because the advective term does not arise in the acoustic wave equation. Without loss of generality, suppose that the domain Ω = [0, L], L > 0. Further, assume there is a single material interface at α (0, L). Throughout this chapter, 4

10 5 the coefficients ρ and β are assumed piecewise constant with ρ L if x α, ρ (x) = otherwise, ρ R (2.3) and β L if x α, β (x) = otherwise, β R (2.4) and 0 < β L, β R, ρ L, ρ R <. We introduce the function space L 2 (Ω) = { v : Ω v2 dx < }, the square integrable functions, and the Hilbert spaces { H 1 (Ω) = v : v, dv } dx L 2 (Ω) (2.5) and H 1 0 (Ω) = { v : v, dv } dx L 2 (Ω), v (0) = v (L) = 0. (2.6) The reason for using the space of functions H 1 is to relax the level of regularity, or smoothness, demanded of the solution. The weak formulation of Eq. (2.1) is then: Find u H 1 0 (Ω) such that L 0 1 u v ρ x x dx + L 0 βuv dx = L 0 fv dx (2.7) for all v H 1 0 (Ω). The boundary term is identically zero because the function space incorporates the boundary conditions. In the case of nonhomogeneous Dirichlet data, the problem can be transformed easily to one with homogeneous Dirichlet boundary conditions provided the boundary function satisfies certain regularity requirements. Thus, the Dirichlet data are assumed homogeneous throughout this section. At this point there is no problem at the interface. As long as we remain in the continuum, the solution space of functions is rich enough to capture any irregularity induced by the coefficient functions; that is, as long as the coefficients remain what one might call reasonable functions residing in nothing worse than L (Ω). Moving to the a finite dimensional approximation space, regularity is relaxed even further by approximating the Hilbert space H 1 0 (Ω) with a polynomial subspace.

11 6 2.2 Standard FEM Approximation Spatial Discretization Suppose the domain Ω is discretized uniformly into segments of length h and call that domain Ω h. The number of segments, called elements, is N = L/h. We choose the size h so that the number of elements is whole. Although Ω = Ω h in this case, we maintain the notation to avoid confusion. Let K j = [x j, x j+1 ] be a representative element in the discretized physical domain Ω h. The approximation space H0 1 is infinite dimensional. We can approximate solutions H0 1 satisfying the weak formulation by a smaller set of functions: the piecewise linear polynomials. Let V h (Ω) = { v C 0 (Ω h ) : j, v [xj,x j+1 ] P 1 }, (2.8) where P 1 is the set of first order polynomials. The restriction of V h (Ω) to the homogeneous Dirichlet boundary conditions is denoted V h,0 (Ω). The standard basis for this set is the so-called hat functions defined by x x j 1, x j 1 x < x j, x j x j 1 φ j (x) = (2.9) x j+1 x, x j x < x j+1. x j+1 x j Although the mesh spacing is uniform, it will be useful in the next section to maintain the notation for nonuniform spacing. The numerical solution u h is expressed in terms of the basis functions in the finite element subspace V h,0. The approximate solution in this nodal basis is u h (x) = N U j φ j (x), j=1 where φ j V h,0 are each C 0 functions with support on K j K j+1 for j = 1... N 1 possessing the delta property φ j (x k ) = δ jk. Substituting u h into the weak formulation Eq. (2.7), and replacing the test functions v with their counterparts in V h,0, we have the approximation to the continuum problem: Find u h V h,0 such that N 1 j=1 N 1 U j (βφ j, φ k ) L2 (K j K j+1 ) + j=1 ( 1 φ j U j ρ x, φ ) k = (f, φ k ) x L2 (Ω h ) L 2 (K j K j+1 ) for all φ k V h,0. The above discretized weak formulation defines a linear system of

12 7 equations. The inner products are conventionally called the mass matrix and stiffness matrix In a more compact form where F is the load vector S kj = M kj = (βφ j, φ k ) L2 (K j K j+1 ) ( 1 φ j ρ x, φ ) k. x L 2 (K j K j+1 ) MU + SU = F (t) F k = (f, φ k ) L2 (Ω h ). The approximation capability of V h,0 depends on the regularity of the solution u to the continuum problem. Provided u is H 2 (Ω), meaning that the coefficients have sufficient regularity to ensure u is twice weakly differentiable, the standard error estimates apply. In the event that the solution has a kink, or a jump in the derivative, the solution cannot be a member of H 2 (Ω). Thus, the standard error estimates predict at best O (h) in L 2 (Ω). This is demonstrated numerically in Section HCE Finite Element Method In this section, the construction of the harmonic coordinate map in 1D is addressed. The effect of its composition with the standard piecewise linear finite element basis is investigated. The name harmonic coordinate enhanced (HCE) finite element method is coined to distinguish this technique from other finite element methods with similar goals. The name ρ-harmonic coordinates is used to reflect the relevance to a specific application: the acoustic wave equation. Throughout, the ρ-harmonic mapping is referred by the abbreviation ρhm and should be read as ρ-harmonic mapping or ρharmonic map depending on the context. References to the ρ-harmonic coordinates themselves will not be abbreviated ρ-harmonic Coordinates The ρhm is the solution to a Laplace problem involving the second-order term from Eq. (2.1) along with a particular choice of boundary conditions. For the sake of generalization as much as possible in 1D, take the domain Ω = [x L, x R ]. Suppose there is an interface at α (x L, x R ). The ρhm in the model problem domain is easily obtained by setting x L = 0 and x R = L. Let F : [x L, x R ] R. We make no assumptions about F other than it solves the

13 8 Algorithm 2.1 Algorithm for Homogenization (Zhang, 2007) 1: Compute the ρ-harmonic map F on fine mesh. 2: Construct multi-scale finite element basis ψ = φ F, compute stiffness matrix S and mass matrix M (ψ is piecewise linear on the fine mesh). boundary value problem ( ) d 1 df = 0, (2.10) dx ρ dx F (x L ) = x L, (2.11) F (x R ) = x R. (2.12) where ρ is as defined in Eq. (2.3). It is interesting to note that F ( Ω) = Ω. The boundary conditions here guarantee that F is a homeomorphism from Ω onto Ω (Alessandrini and Nesi, 2001). The exact solution to this auxiliary problem is x L + ρ L M (x x L), x α F (x) = x L + ρ L M (α x L) + ρ R M (x α), otherwise, (2.13) where M = (ρ L (α x L ) + ρ R (x R α)) / (x R x L ) is the average mass density for the domain. In this case, we are dealing with linear mass densities. Notice that the densities cancel and leave only coordinates satisfying dimensionality. The application of ρ-harmonic coordinates in the context of upscaling follows Algorithm 2.1. Basis functions are constructed in the regular grid and composed with the fine scale harmonic map to produce basis functions that exist in the coarse grid but contain fine scale information. An unfortunate side effect of this technique is that the basis functions now reside on some distorted support. Since keeping a regular solution grid is desirable, an alternative procedure is proposed. The HCE basis construction deviates from the core approach in Algorithm 2.1 in that the basis is defined on the image of the regular grid under F rather than the physical grid. The amendment presented here adds one and modifies another step in the upscaling procedure. Instead of simply composing the basis functions on the coarse mesh with ρhm, the coarse grid should first be mapped under ρhm to produce an alternative grid, which is called the harmonic coordinate grid. The piecewise linear basis functions are defined on this distorted grid. By composition with ρhm, the basis functions retain the desired fine scale information and regain the regular grid support. It is shown in the next section that this method results in the initial regular grid being recovered for exploitation. With the new steps indicated in red, the modified

14 9 approach is Algorithm 2.2. Such a method has been presented in the case of simplicial elements (Zhang, 2007) as an alternative to the standard homogenization algorithm and is revisited in Chapter 4. The basis functions for the 1D case are constructed explicitly in the next section. Algorithm 2.2 Modified Homogenization Algorithm 1: Compute the ρ-harmonic map F on fine mesh. 2: Obtain harmonic coordinate grid F (Ω h ). 3: Construct piecewise linear basis {ψ} on F (Ω h ). 4: Construct HCE finite element basis φ = ψ F, compute stiffness matrix S and mass matrix M HCE Finite Element Basis Let Ω h be a uniform partition of Ω = [0, L] as stated in the model problem. Let F be given by Eq. (2.13). Denote the harmonic coordinate grid generated from Ω h under F by F (Ω h ) and the grid points y j = F (x j ). The mapping F is a bijection from Ω h onto Ω h (a homeomorphism). Since F leaves the boundary unchanged, only the interior grid points are redistributed. Denote the image of the interface point in the harmonic grid as η = F (α). Following Algorithm 2.2, the piecewise linear basis {ψ j } on F (Ω h ) is constructed as follows. Consider two elements F (K) j,f (K) j+1 in the harmonic coordinate grid. Let j = 1,..., N be an index over the number of elements F (K) j in the domain. Then ψ j : F (K j K j+1 ) [0, 1] and is defined as y y j 1, y j 1 y < y j, y j y j 1 ψ j (y) = (2.14) y j+1 y, y j y < y j+1. y j+1 y j An illustration of this basis in the harmonic grid is shown in Fig We ignore the domain boundary since the particular problem has homogeneous Dirichlet boundary conditions specified. At the material interface, the length of the support intervals changes, so there is a sort of stretching of the basis functions about the point η (see Fig. 2.1). By composing φ j with ρhm, new basis functions are found that take data from the physical grid Ω h and incorporate material changes. Clearly, only changes in ρ are contained in this mapping; the coefficient β is not incorporated. The new basis functions are defined as φ j (x) = ψ j (F (x)). (2.15) That the new basis is indeed locally modified is easily checked.

15 10 ψ m ψ m+1 η 0 y m 2 y m 1 y m y m+1 y m+2 L Figure 2.1: Cartoon of the basis functions in the harmonic coordinate grid. Suppose spt φ j < α. Then ρ = ρ L is constant across both elements. Compute φ j by substituting F into Eq. (2.14). Restricting to K j φ (x) Kj = F (x) y j 1 y j y j 1. (2.16) The denominator is actually y j y j 1 = F (x j ) F (x j 1 ) which reduces to ( F (x j ) F (x j 1 ) = x L + ρ ) ( L M (x j x L ) x L + ρ ) L M (x j 1 x L ) (2.17) = ρ L M (x j x j 1 ) (2.18) The numerator is simplified in the same way to find ( F (x) F (x j 1 ) = x L + ρ ) ( L M (x x L) x L + ρ ) L M (x j 1 x L ) (2.19) = ρ L M (x x j 1). (2.20) Substituting the expressions Eq. (2.18) and Eq. (2.20) into Eq. (2.16) we find φ j (x) Kj = x x j 1 x j x j 1.

16 11 ψ m F ψ m+1 F 0 x m 2 x m 1 x m ξ x m+1 x m+2 L Figure 2.2: Cartoon of the basis functions mapped from the harmonic coordinate grid back to the regular grid. The kinks occur where the flux jump is forced to zero when the density is discontinuous. This introduces regularity into the solution. Similarly, φ (x) Kj+1 = x j+1 x x j+1 x j since F (x j+1 ) F (x j ) = ρ L M (x j+1 x j ) and F (x j+1 ) F (x) = ρ L M (x j+1 x). Thus, when spt φ j < α the new basis reduces to the usual piecewise linear basis. By simply following the same approach as above, it is easy to verify that the basis has the form x x j 1, x j 1 x < x j, x j x j 1 φ j (x) = x j+1 x, x j x < x j+1, x j+1 x j when spt φ j > α. If α spt φ j, then the basis functions have kinks as in Figure 2.2. These functions are computed using Eq. (2.16) where the coefficient ρ cannot be eliminated. Obviously, when ρ L = ρ R, these functions reduce to the standard piecewise linear basis regardless of the coefficient β Equivalence of IFEM and HCE-FEM in 1D We present a simple verification that the bases obtained by the procedure in Algorithm 2.2 and IFEM are equivalent. The IFEM basis construction is well established (Li, 1998; Li and Ito, 2006).

17 12 Theorem (IFEM-Owhadi Basis Equivalence). The basis obtained by Algorithm 2.2 is equivalent to the IFEM basis in 1D. Proof. This proof proceeds on two fronts. We first compute the IFEM basis for a given discretization. The element containing the interface is the only interesting part. Without loss of generality, use the uniform grid x j = jh for j = 0,..., N with x 0 = 0 and x N = 1. Use the standard P 1 Lagrange basis, defined in Eq. (2.9). Suppose the interface α (x j, x j+1 ) so it is not coincident with a grid point. Let be defined as in Eq. (2.3). In the presence of an interface, the IFEM basis is defined by the δ-property and the jump conditions [ ] [φ j ] x=α = 0, ρ 1 φ j = 0 x=α restricting the basis to be C 0 and placing a physical constraint on the flux. Within (x j, x j+1 ) assume a form { φ L j (x) = D L (x j x) + 1, x [x j, α), φ j (x) = φ R j (x) = D R (x j+1 x) x (α, x j+1 ) where L/R indicates left or right of the interface. Clearly, φ L j (x j ) = 1, and φ R j (x j+1 ) = 0. By applying the continuity conditions, the unknown coefficients are ( ) 1 D = h + ρr ρ L (x L j+1 α), 1 D R = ρ L ρ R D L. After substitution and a little simplification, the IFEM basis function is x x j, x j x x j+1, h ρ L (x j x) IF EM φj (x) = ρ L h (ρ R ρ L ) (x j+1 α) + 1, x j x α, ρ R (x j+1 x) ρ L h (ρ R ρ L ) (x j+1 α), α x x j+1. ρ L The other basis function whose support contains the interface is computed in exactly the same way. Now we consider the HCE-FEM basis. Since we have already shown that the basis functions are standard away from the interface, we consider only the part of the basis

18 13 involving [x j, x j+1 ]. By Eq. (2.13) and Eq. (2.16), the basis function is OF EM φj (x) = ρ L (x j x) ρ L h (ρ R ρ L ) (x j+1 α) + 1, x j x α, ρ R (x j+1 x) ρ L h (ρ R ρ L ) (x j+1 α), Verifying the other basis function follows the same argument Stiffness Matrix α x x j+1. The entries of the stiffness matrix depend on derivatives of the basis functions. Since φ j are composite functions, the chain rule applies. The derivative is dφ j dx = dψ j df df dx (2.21) The ρhm is piecewise linear, so the derivative is piecewise constant yielding ρ L df if x α, dx = M ρ R M if x > α. (2.22) The basis functions ψ j, which are defined on the harmonic grid, are piecewise linear hat functions; they too have piecewise constant derivatives 1 dψ, y j 1 y y j, j dy = y j y j 1 1 (2.23), y j y y j+1. y j+1 y j The derivative of φ j has different cases depending on whether the support contains any material discontinuity. Let F (α) = η. In the case α > spt φ j, the mapped interface η > spt ψ j, yielding dφ j dx ρ L M (y j y j 1 ), y j 1 F (x) y j, (α > x) = ρ L M (y j+1 y j ), y j F (x) y j+1. (2.24)

19 14 Similarly, when α < spt φ j, η < spt ψ j and ρ R dφ j M (y j y j 1 ), y j 1 F (x) y j, (α < x) = dx ρ R M (y j+1 y j ), y j F (x) y j+1. (2.25) These functions reduce to the standard piecewise linear basis in the physical grid Ω h. When α spt φ j, there are two possible scenarios: η [y j 1, y j ] or η [y j, y j+1 ]. The first case gives ρ L M (y j y j 1 ), y j 1 F (x) η, dφ j dx (x) = while in the second case dφ j dx (x) = ρ R M (y j y j 1 ), ρ R M (y j+1 y j ) ρ L η < F (x) y j y j F (x) y j+1 (2.26) M (y j y j 1 ), y j 1 F (x) y j, ρ L M (y j+1 y j ), y j F (x) η. (2.27) ρ R η < F (x) y j+1 M (y j+1 y j ) The main diagonal entries computed directly from the definition of the composite basis functions are ( ) 1 1 ρ S jj = L (x j x j 1 ) + 1, α > spt φ j ( (x j+1 x j ) ) (2.28) 1 1 (x j x j 1 ) + 1, α < spt φ j. (x j+1 x j ) ρ R Exactly two main diagonal terms will contain the discontinuity. Assume that the discontinuity occurs in element 1 < (k + 1) < N. Then φ k and φ k+1 are affected. So, and S kk = ρ L (x k x k 1 ) M 2 (y k y k 1 ) 2 + ρ L (α x k ) M 2 (y k+1 y k ) 2 + ρ R (x k+1 α) M 2 (y k+1 y k ) 2 (2.29) S k+1,k+1 = ρ R (x k+2 x k+1 ) M 2 (y k+2 y k+1 ) 2 + ρ R (x k+1 α) M 2 (y k+1 y k ) 2 + ρ L (α x k ) M 2 (y k+1 y k ) 2 (2.30)

20 15 Of the upper and lower diagonal terms, it is obvious that only two of these can be affected. For the discontinuity in element (k + 1), S k,k+1 = ρ L x k α M 2 (y k+1 y k ) 2 + ρ R α x k+1 M 2 (y k+1 y k ) 2. (2.31) Symmetry gives S k,k+1 = S k+1,k. The upper and lower diagonal entries for the case where the support does not include α are easily obtained from the terms above Mass Matrix First consider an arbitrary element K k and its next neighbor K k+1. Provided for all x spt φ k, x < α, we have M kk = 1 xk+1 φ k (x) 2 dx. (2.32) κ L x k 1 Note that the material property function κ is piecewise constant. Recall that φ k (x) = ψ k (F (x)). Then M kk = 1 xk+1 ψ k (F (x)) 2 dx (2.33) κ L x k 1 Since ψ k is piecewise linear, separate the integral and substitute the expressions for each part of the support. As was shown in Section 2.3.2, the basis functions φ k collapse to the usual hat functions when the support does not contain a material change. A simple calculation reveals the diagonal mass matrix entries, when α / spt φ k, are 1 (x k+1 x k 1 ), α > spt φ k, M kk = 3κ L 1 (2.34) (x k+1 x k 1 ), α < spt φ k, 3κ R where the material property is the only thing that changes. The mass matrix entries are affected in exactly the same way as the stiffness matrix entries when the support of the basis functions include the discontinuity. Suppose the discontinuity occurs in element 1 < (k + 1) < N. Then basis functions φ k and φ k+1 are affected. A tedious but straight-forward integration reveals the mass matrix diagonal entry associated with φ k is M kk = 1 (x k x k 1 ) + 1 µ k+1 ( ) 1 b ρ 2 1 (x 3κ 0 3κ 0 ρ 0 3κ 1 µ 2 k+1 α) 3, (2.35) k+1 where µ k+1 = ρ R (x k+1 α) + ρ L (α x k ),

21 16 and A similar calculation yields b = 1 + (ρ L (x k α)) /µ k+1. M k+1,k+1 = ρ 2 0 3µ 2 k+1 κ 0 (α x k ) 3 + µ k+1 3ρ 1 κ 1 ( 1 a 3 ) + 1 3κ 1 (x k+2 x k+1 ), (2.36) where µ k+1 is as previously defined, and a = 1 + ρ 1 (α x k+1 ) /µ k+1. The sub- and superdiagonal entries are computed from M k,k+1 = xk+1 1 x k α = 1 κ L κ φ k (x) φ k+1 (x) dx x k φ k φ k+1 dx + 1 κ R xk+1 α φ k φ k+1 dx (2.37) After a simple calculation, the integral reduces to where µ k+1 is as defined above, and M k,k+1 = 1 κ L µ k+1 6ρ L ( 3c 2 2c 3) + 1 κ R µ k+1 6ρ R ( 3d 2 2d 3), (2.38) c = ρ 0 µ k+1 (α x k ), d = ρ 1 µ k+1 (x k+1 α). Entries that do not contain an interface are easily determined from this expression. By symmetry, M k,k+1 = M k+1,k. 2.4 Numerical Examples The following examples demonstrate computationally that the stiffness and mass matrices constructed above provide an approximation that exhibits optimal convergence.

22 17 Example 1 Consider the ordinary differential equation ( d σ du ) = 12x 2, dx dx u (0) = 0, u (1) = 0, where σ is piecewise constant. Assume there is an interface at 0 < α < 1 and define { σ L, x < α, σ (x) = σ R, x α The exact solution for this problem is where x 4 + cx, x < α, u (x) = σl x 4 + cα + α4 σr ( 1 σ L 1 σ R ( 1 c = α 3 σ 1 ) 1 R σ L ασ. R ), x α, Successive refinements reveal the convergence of the HCE-FEM method is O (h 2 ) while standard FEM is O (h). The rates and errors in the L 2 -norm are presented in Tables 2.1(a-d). Example 2 As with the stiffness matrix, we choose a simple test problem to verify that the mass matrix is correct. This is by no means an exhaustive test, but it should be enough to argue that the assembled matrix is correct. Consider the differential equation d dx ( σ du dx ) + ωu = x, u (0) = 0, u (1) = 0,

23 18 Table 2.1: L 2 error and average rates for the stiffness matrix test problem. N FEM IFEM e e e e e e e e e e (a) σ L = 1, σ R = 10 N FEM IFEM e e e e e e e e e e (c) σ L = 10, σ R = 1 N FEM IFEM e e e e e e e e e e (b) σ L = 1, σ R = 100 N FEM IFEM e e e e e e e e e e (d) σ L = 100, σ R = 1

24 19 Again σ is piecewise constant. Likewise, we choose ω piecewise constant. Assume there is an interface at 0 < α < 1 and define { σ L, x < α, σ (x) = σ R, x α, and ω (x) = { ω L, x < α, ω R, x α. The exact solution for this problem is A sin ( γ L x ) x u (x) = ω, x < α, L B sin ( γ R x ) + C sin ( γ R x ) x ω, x > α, R ω L ω R σ R. The coefficients are easily determined by impos- where γ L = and γ R = σ L ing continuity and the flux condition at the interface along with the final boundary condition. A reasonable first test is σ L = σ R = 1 so that the stiffness matrix is the standard finite element stiffness matrix. Any contribution with a ratio of ω L /ω R 1 should reveal limitations in the method for the case of constant σ. We choose ratios ω L /ω R = 1/10, 1/100, 10, 100. FEM and HCE-FEM are equivalent for constant σ, and both methods perform with surprising accuracy given the contrast in ω. Results for this test are shown in Tables 2.2(a-d). The harmonic coordinate mapping only comes into play when the density term is not constant. Therefore, take σ L /σ R = 1/10 as in the stiffness matrix tests to check the behavior of these methods when the contrasts are high. From Table 2.3 the convergence was solidly second order for the HCE-FEM and first order for standard finite elements. This demonstrates the well-known deficiency in FEM for such problems.

25 20 Table 2.2: L 2 error and average rates for the mass matrix test problem with σ L /σ R = 1. N FEM IFEM e e e e e e e e e e (a) ω L /ω R = 1 10 N FEM IFEM e e e e e e e e e e (c) ω L /ω R = 10.0 N FEM IFEM e e e e e e e e e e (b) ω L /ω R = N FEM IFEM e e e e e e e e e e (d) ω L /ω R = 100.0

26 21 Table 2.3: L 2 error and average rates for the mass matrix test problem with σ L /σ R = 1/10. N FEM IFEM e e e e e e e e e e (a) ω L /ω R = 1 10 N FEM IFEM e e e e e e e e e e (c) ω L /ω R = 10.0 N FEM IFEM e e e e e e e e e e (b) ω L /ω R = N FEM IFEM e e e e e e e e e e (d) ω L /ω R = 100.0

27 Chapter 3 HCE-FEM Applied to 1D Acoustic Wave Equation 3.1 Model Problem Let Ω R be the interval Ω = [0, L]. Suppose there is a material interface at α (0, L). Consider the homogeneous one-dimensional acoustic wave equation 1 2 u κ t ( ) 1 u = 0 in Ω [0, T ] (3.1) 2 x ρ x where κ, ρ L (Ω) and u : Ω [0, T ] R is the unknown solution u (x, t). In this case, the solution is the acoustic pressure. Prescribe the initial conditions u (x, t = 0) = f (x), u (x, t = 0) = g (x), x Ω, (3.2) t with f, g smooth (better than H 2 (Ω)), and apply homogeneous Dirichlet boundary conditions u (x, t) = 0, x Ω, t [0, T ]. (3.3) As with the elliptic case in the previous chapter, the same problem with nonhomogeneous Dirichlet boundary conditions can be easily transformed to Eq. (3.1). Let the denisty and bulk modulus be piecewise constant functions defined by ρ L if x α, ρ (x) = otherwise, ρ R 22

28 23 and κ L if x α, κ (x) = otherwise, κ R with 0 < κ L, κ R, ρ L, ρ R <. The wave speed c is governed by the material properties and expressed by the well-known relation κ c = ρ. (3.4) The weak formulation of Eq. (3.1): Find u (, t) H 1 0 (Ω) for t [0, T ] such that L 0 1 d 2 u L κ dt v dx u v dx = 0, (3.5) ρ x x for all v H 1 0 (Ω). The boundary integral arising from the application of Green s identity is zero because the function space incorporates the boundary conditions. Since the coefficients are piecewise constant time-independent functions, this weak formulation closely resembles the result in Eq. (2.7). The solution u can be considered a time-dependent function with spatial variation in H 1 0 (Ω). This motivates the definition of the space of functions which are C k in time and H 1 0 in space. C k ( [0, T ], H 1 0 (Ω) ), (3.6) 3.2 Discretization Spatial Approximation Let Ω be discretized uniformly. Take approximation space for the spatial component V h,0, where V h is defined in Eq. (2.8). Define the approximation space Suppose u h Vh,0 t defined as V t h,0 = C 2 ([0, T ], V h,0 ) u h (x, t) = N 1 j=1 U j (t) φ h (x)

29 24 Substituting this into Eq. (3.5) defines a system of ordinary second order differential equations. The approximation problem is: Find u h Vh,0 t such that N 1 j=1 d 2 U j dt 2 spt φ j 1 κ φ jφ k dx + N 1 j=1 1 φ j φ k U j (t) dx = 0, (3.7) spt φ j ρ x x for all φ k V h,0. Note that the integrals correspond to the mass and stiffness matrices computed in the previous chapter. Let u (t) denote the time-dependent vector of coefficients for u h. Then Eq. (3.7) is more compactly written as Time Approximation M d2 u + Su = 0. dt2 Since the mass matrix is independent of time and invertible, d 2 u dt 2 + M 1 Su = 0. The second order time derivative is approximated as d 2 u dt 2 The resulting leapfrog scheme is un+1 2u n + u n 1 t 2. u n+1 2u n + u n 1 t 2 + M 1 Su n = 0. Naturally, the time interval must be chosen to satisfy the CFL condition to ensure stability. 3.3 Numerical Examples The approximation capability of this method is tested in this section. The PDE to solve is Eq. (3.1). Let R (t) = ( 1 2 (πf 0 (t t 0 )) 2) e (πf 0(t t 0 )) 2

30 25 which is the Ricker wavelet (Ricker, 1945) with central frequency f 0 = 10 Hz. The exact solution written in terms of the Ricker wavelet R (t x/c L ) c Lρ L c R ρ R R (t + x/c L ) for x α, u (x, t) = c L ρ L + c R ρ R 2c R ρ R R (t x/c L ) for x > α. c L ρ L + c R ρ R The time trace of the solution at x = 500 m is interpolated to a fine grid by cubic spline. The reference solution, which is know exactly, is evaluated on the same fine grid. To compare the methods, the relative RMS error is computed in the l 2 -norm by RRMS(u, u h ) = u u h l2 u l2. All traces are measured for t max = 600 ms on three meshes having element size h = 5 m, 2.5 m, 1.25 m. The time step is chosen to satisfy stability. Assume throughout that the domain is 0 x 2500 m and the interface is located at α = m. Initially, assume ρ L = 2100 kg/m 3, ρ R = 2300 kg/m 3, c L = 2.3 m/ms, and c R = 3.0 m/ms corresponding to the data used in (Symes and Vdovina, 2008); their application to the first order acoustic system provides relevant data to show a similar limitation in finite elements and provide a starting point for further testing. The bulk modulus κ can be obtained by using Eq. (3.4). In Table 3.1, both the standard FEM and HCE-FEM perform well. The L 2 error is computed between the exact and numerical solution of the reflected wave interpolated on the spatial grid. The transmitted portion of the wave is not considered. The relative RMS error is computed as described above. These data reveal that standard FEM is a reasonable choice for both constant and low contrast density. Assume c L = 2.3 m/ms, c R = 3.0 m/ms as before. With ρ L = 230 kg/m 3 and ρ R = 2300 kg/m 3, the performance disparity between standard FEM and HCE-FEM is clear, see Table 3.2. Standard FEM is equivalent to finite difference on the regular grid, so it is not surprising that the large density contrast with an misaligned interface would produce first order accuracy. The RMS error is comparable to the error seen for the staggered grid finite difference scheme with h = 5 m.

31 26 Table 3.1: Error rate for acoustic wave equation test with data from Symes and Vdovina (2008) with and without mass lumping. The L 2 -error is measured in the spatial grid and only involves the reflected wave. FEM HCE-FEM h L 2 -error RRMS (%) L 2 -error RRMS (%) e e e e e e (a) ρ L = 2100 kg/m 3, ρ R = 2300 kg/m 3 FEM HCE-FEM h L 2 -error RRMS (%) L 2 -error RRMS (%) e e e e e e (b) ρ L = 2100 kg/m 3, ρ R = 2300 kg/m 3 with mass lumping Table 3.2: Relative RMS error (%) for acoustic wave equation test with wave speeds from Symes and Vdovina (2008), high density contrast, and with and without mass lumping. h FEM HCE-FEM (a) ρ L = 2300 kg/m 3, ρ R = 230 kg/m 3. h FEM HCE-FEM (b) ρ L = 2300 kg/m 3, ρ R = 230 kg/m 3 with mass lumping.

32 Chapter 4 HCE Finite Elements for 2D Elliptic Problems The equivalence between IFEM and HCE-FEM is confined to 1D. Owhadi and Zhang developed a theory that clearly establishes optimal rates. Yet, all numerical results to date exhibit little better than superlinear rates of convergence. Accurately computing the inner products in the discretized weak formulation is necessary to overcome this limitation. The nature of the HCE-FEM construction leads to nonconforming finite elements if the support of the basis functions is not faithfully represented. We present a method to accurately integrate the composite basis functions using the fine grid computation of the harmonic map. In this chapter I follow the same approach as in the 1D case where the model problem is described and cast into a weak formulation. Fundamental notation and tools are provided, which lead immediately to the standard discretization for the finite element method. Once established, this foundation provides a starting point to develop the HCE-FEM in 2D. The integration algorithm is defined. 4.1 Model Problem Let Ω R 2 be a simple closed domain with boundary Ω. Unlike the 1D case, the interface may interact with the domain boundary in 2D and higher. Suppose there is a material interface described by a curve Γ Ω. Consider the scalar boundary value problem ( ) 1 ρ u + η u + βu = f in Ω (4.1) u = 0 x Ω, (4.2) 27

33 28 where ρ [L (Ω)] 2,2, η [L (Ω)] 2, β L (Ω), f L 2 (Ω), and u : Ω R is the unknown solution. Throughout, ρ ij > 0, which means the differential operator in Eq. (4.1) satisfies the definition of ellipticity. Define the function spaces L 2 (Ω) = {u : u 2 < } H 1 (Ω) = {u : u, u L 2 (Ω)} Ω and H 1 0 (Ω) = { u : u H 1 (Ω) and u = 0 on Ω } Multiply Eq. (4.1) by v H0 1 (Ω) and integrate over the domain Ω. The weak form is: Find u H0 1 (Ω) such that v 1 u + v (η u) + βuv = fv, (4.3) ρ Ω for all v H 1 0 (Ω). The boundary term corresponding to Ω is eliminated upon application of Green s identity since the test and approximation function spaces include the homogeneous boundary condition. The weak form obtained here corresponds to the interface problem when there is no flux at the interface. Explicitly indicating an interface with regard to the coefficient matrix 1/ρ is not necessary in the continuum. The acoustic wave equation in divergence form has no term corresponding to the coefficient η, so let η = 0 from this point forward. As defined, the bilinear form associated with the model problem is coercive. The problem is well-posed. Ω 4.2 Preliminaries Notational and Semantic Conventions The following conventions of Ern and Guermond (2004) are used. When referring to the functions used to approximate geometrical objects, such as the approximation functions for reference mappings from reference to physical elements, the functions are called geometric reference shape functions. There are two sets of geometric reference shape functions. The first set is denoted by { ψ 1, ψ 2,..., ψ ngeo } with n sh indicating the number of geometric reference shape functions. The second set is { ϕ 1, ϕ 2,..., ϕ ngsh } with n gsh indicating the number of degrees of freedom. All reference functions and elements are indicated by a hat. The standard reference element is K. The shape functions defined on the reference element used to approximate the solution are called local shape functions on K. These functions are represented by { θ 1, θ 2,..., θ nsh } and

34 29 are associated with the geometric reference shape functions ψ. The local shape functions mapped to the physical domain are denoted by θ n,k and referred to as global shape functions; the K indicates the particular element. The distinction between local shape functions and the geometric reference shape functions is important when the elements are distorted. Computing the derivatives of the global shape functions ultimately involves derivatives of local shape functions scaled by the Jacobian matrix, which is define by [Df] k,l = f k x l. The term Jacobian is reserved for the determinant The form J f = det Df. J f (x) = det D x f is used when the dependent variable must be shown explicitly. element mappings, the element placeholder is used: In the context of J K (ˆx) = det Dˆx T K. Often the dependent variable is obvious from the context and will be exploited to reduce notation complexity Q 1 Lagrange Finite Elements The reference Q 1 element is defined as the biunit square K, a set of the local shape functions { θ 1, θ 2,..., θ nsh }, and local degrees of freedom numbering n sh = 4. Physically, the reference quadrilateral K is the square defined by the convex hull of Q = {[ 1, 1], [1, 1], [1, 1], [ 1, 1]}. The shape functions are interpolatory in that θ j ( x k ) = δ jk where x k Q. The shape functions associated with Q 1 are the bilinear functions θ 1 (ξ, η) = 1 4 (1 ξ) (1 η), θ2 (ξ, η) = 1 (1 + ξ) (1 η), 4 θ 3 (ξ, η) = 1 4 (1 + ξ) (1 + η), θ4 (ξ, η) = 1 (1 ξ) (1 + η), 4 where 1 ξ, η 1. The argument of the shape functions shall interchangeably be x and the pair (ξ, η) when explicitness is required. For the purposes of this work, the geometric shape functions ψ and local shape functions θ are the same. Under some circumstances it is useful to keep them separated in software, so the notation is maintained in the interest of generality.

35 θ 1 θ 2 θ 3 θ 4 Figure 4.1: The Q 1 Lagrange basis functions. Note that the each function is unity at a single node Reference Map Let the vertices for the reference element K be Q. Suppose the vertices of a quadrilateral element in the physical mesh K Ω h are K = {a 1, a 2,..., a ngeo }, where n geo = 4 for bilinear quadrilateral elements. The mapping from the reference element to the physical element is easily constructed in terms of the bilinear geometric shape functions { ψ 1, ψ 2,..., ψ ngeo } by n geo T K ( x) = a j ψj ( x) j=1 where x K (see Fig. 4.2). Provided the physical elements are no worse than parallelograms, the mapping T K is affine. Thus, DT K is constant over the element K. For general quadrilaterals, the Jacobian matrix must be computed at each point. 4.3 Harmonic Grid The harmonic map corresponding to the model problem is the solution to ( ) 1 ρ F = 0 in Ω (4.4) F = x x Ω, (4.5) where F = (F 1, F 2 ) with F j : Ω R for each j, and the boundary condition F j = x j for all x Ω. The problem is well-posed. The solution is a homeomorphism on Ω with F : Ω Ω. Assume that the solution to this boundary value problem is known exactly, or at least known as precisely as is required, whatever that might mean for the problem at hand. A freely available solver using multigrid, such as PLTMG (??), or a combi-

36 31 T K ( 1, 1) η (1, 1) a 4 a ξ x 2 a 1 a ( 1, 1) (1, 1) x 1 Reference Element K Physical Element K Figure 4.2: The bilinear reference map T K : K K. Note the order of the nodes is preserved. nation of adaptive mesh refinement and multigrid, such as AMRPoisson (?) can be used to obtain a highly accurate solution to this problem. Regardless of the method, the harmonic coordinate map is assumed arbitrarily accurate. Let the regular quadrilateral mesh of the domain Ω be denoted Ω h. Suppose the nodes of the mesh Ω h are ω k. Under the mapping F, the nodes {F (ω k )} k define a new mesh F (Ω h ) called the harmonic grid. The true image of the regular grid elements under F have curved and even piecewise continuous edges. Instead, only the nodes are mapped and the element edges are assumed straight in F (Ω h ). Thus, an approximate harmonic grid made of quadrilateral elements is obtained. For the remaining sections, F (K) will denote the quadrilateral approximation in the harmonic grid of a representative element K Ω h D HCE-FEM Composed Basis In principle, the basis functions for 2D are constructed in exactly the same manner as those in 1D. The process is slightly more involved for two reasons. First, an explicit form for the harmonic map is not known. The basis functions are no longer representable as a simple algebraic expression. The other issue is the harmonic map gives general quadrilateral elements where the global shape functions must be defined using a reference map. Let K be a representative element from Ω h, and F (K) the quadrilateral image of K. Define the bilinear global shape functions {θ j,k } n sh j=1 on F (K). By the composition method used in 1D, the the global shape functions corresponding to θ j,k in the