Mixed Hybrid Finite Element Method: an introduction
|
|
- Osborn Robertson
- 6 years ago
- Views:
Transcription
1 Mixed Hybrid Finite Element Method: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato in Scienze dell Ingegneria Civile e Ambientale Corso di Metodi Numerici marzo 2008
2 Outline Some observation on the Finite Element method The Mixed Finite Element method The Mixed Hybrid Finite Element method Some properties of the method Superconvergence results Numerical implementation Numerical results A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
3 About the Finite Element method It is well known that the classical Finite Element (FE) method minimizes the so called energy functional) in a well chosen space of admissible functions W inf J(w). w W If the functional J( ) is differentiable, the minimum (whenever it exists) will be characterized by a variational equation. The FE method is based on a few simple ideas: the domain Ω R d, d = 2, 3, in which the problem is posed, is partitioned into a set of simple subdomains, called elements. These elements may be triangles, quadrilaterals, tetrahedra. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
4 A space W of functions defined on Ω is then approximated by simple functions, defined on each element with suitable matching conditions at interfaces. Simple functions are commonly polynomials or functions obtained from polynomials by a change of variables. A FE method can only be considered in relation with a variational principle and a functional space. Changing the variational principle and the space in which it is posed leads to a different FE approximation, even if the solution for the continuous problem can remain the same. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
5 The Mixed Finite Element method In the Mixed Finite Element (MFE) method two different Finite Element spaces are used. Let us consider the simple elliptic equation: (D c) = f c = 0 in Ω on Ω where D = D(x) is the dispersion tensor, and c represents the concentration. Introducing the dispersive flux G, we can write the Fick law, G = D c. It could be desiderable to approximate G and c simultaneously using a different Finite Element space for each variable. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
6 With this purpose the elliptic problem is decomposed into a first order system as follows: G + D c = 0 G = f c = 0 in Ω in Ω on Ω The first of the above equations can be written as D 1 G + c = 0 in Ω. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
7 Multiplying by test functions and integrating by parts we obtain the following weak formulation D 1 G w d c w d = 0 w H(div, Ω) Ω Ω ψ G d = f ψ d ψ L 2 (Ω) Ω Ω where H(div, Ω) = { w L 2 (Ω) d : w L 2 (Ω)} Hilbert space The weak formulation involves the divergence of the solution and test functions and not arbitrary first derivatives. Thus we work with the space H(div, Ω) formed by piecewise polynomial vector functions with continuous normal component. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
8 In order to define finite element approximations to the solution ( G, c), we need to have finite element subspaces of H(div, Ω) and L 2 (Ω). Let T = {T l } m l=1 be a triangulation of Ω, i.e. Ω = T l T T l with diameter h. The triangulation is admissibile if the intersection of two triangles is either empty, or a vertex, or a complete side. Thus, we have to construct piecewise polynomials spaces W h and Ψ h associated with T l such that W h H(div, Ω) and Ψ h L 2 (Ω). A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
9 The MFE approximation ( G h, c h ) W h Ψ h is defined by Ω Ω D 1 Gh w d c h w d = 0 w W h Ω ψ Gh d = f ψ d ψ Ψ h. Ω In order to have stability and convergence W h and Ψ h can not be chosen arbitrarily but they have to be related. We assume that W h = Ψ h. Let π 2G be the L 2 - projection of G into Ψ h. The following equality must hold: ( G π 2G)ψ d = 0 G H 1 (Ω) d, ψ Ψ h Ω where H 1 (Ω) is the Sobolev space (H 1 (Ω) = {v L 2 (Ω) : D 1 v L 2 (Ω) D 1 v partial derivative}) A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
10 We can state the following theorem: Theorem The problem of finding a pair of functions ( G, c) H(div, Ω) L 2 (Ω) such that the weak form of the elliptic problem holds has a unique solution. In addition, c is the solution of the elliptic problem and G = D c. Proof is omitted. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
11 Numerical implementation of the MFE method A MFE scheme is given by defining the spaces W h and Ψ h approximating W H(div, Ω) and Ψ L 2 (Ω) respectively. We use the Raviart-Thomas spaces defined on a generic element T l Ω as RT k = (P k ) d + xp k where k is an integer 0 and P k is the space of polynomials of degree k. The dimension of RT k is given by { (k + 1)(k + 3) for d = 2 dimrt k = 1(k + 1)(k + 2)(k + 4) for d = 3. 2 A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
12 Lemma For w h RT k the following relations hold: w h P k w h n Tl R k. where R k is the polynomial space defined on the edges e j of each element T l : R k = {φ : φ L 2 ( T l ), φ ej P k, e j }. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
13 We consider the Raviart-Thomas space of degree zero, whereby the functions G and c can be approximated by: G G = c c = m g l w l l=1 m c l ψ l where w l and ψ l are vector and scalar basis functions. Since we are considering the RT 0 spaces, w l are first order polynomial of the type: w l = l=1 ( ax + b ay + c while ψ l are P 0 polynomials equal to one on element T l and zero elsewhere. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24 ),
14 Lemma Given a triangle T l with edges e j, j = 1, 2, 3, the following relationships hold for w l W h (T l ): w l P 0 w l n j l R 0 j = 1, 2, 3 where n j l is the outward normal to edge e j of T l and R 0 is the polynomial space defined on edge e j of T l as R 0 = {φ : φ L 2 ( T l ), φ ej P 0, e j }. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
15 Proof. The first relation is very simple to prove: w l = 2a P 0. The second relation says that the restriction of w l to each of the edges of T l coincides with a polynomial of degree zero, i.e. a constant. Let T l be an element with nodes P 1 = (x 1, y 1 ), P 2 = (x 2, y 2 ), P 3 = (x 3, y 3 ). The normal to the edge e 1 = P 1 P 2 is given by: n 1 l = ( y2 y 1 x 1 x 2 ). A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
16 Continued. Noting that the line passing through the points P 1 and P 2 has equation: x(y 2 y 1 ) + y(x 1 x 2 ) = x 1 (y 2 y 1 ) + y 1 (x 1 x 2 ), the inner product w l n 1 l can be written as w l n 1 l = (ax + b)(y 2 y 1 ) + (ay + c)(x 1 x 2 ) = a[x 1 (y 2 y 1 ) + y 1 (x 1 x 2 )] + b(y 2 y 1 ) + c(x 1 x 2 ) = const. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
17 About RT0 basis functions On each element T l we choice the RT0 basis functions w l i (i = 1, 2, 3) of the following form: ( ) a w l i i = l x + bl i al iy + ci l It is satisfied the following property (Kronecker property): { w j 1 if j = i l n i l ds = δ ij = 0 otherwise e j where n i l is the outward normal to e i. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
18 We can now derive the coefficients a i l, bi l and c i l coefficients in the following way (for sake of simplicity, we will omit the sub- super- scripts l, i). Given the triangle T l with edges e 1, e 2, e 3 and nodes P 1 = (x 1, y 1 ), P 2 = (x 2, y 2 ), P 3 = (x 3, y 3 ), the normal to edge e 1 = P 1 P 2 is derived by: x y 1 x 1 y 1 1 x 2 y 2 1 = 0 that is α(x x 1 ) + β(y y 1 ) = 0, where α = y 2 y 1, β = x 1 x 2. The normalized components of the normal to e 1 are then (with no consideration about inner or outward normal) : n x = α α2 + β 2, n y = β α2 + β 2. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
19 When imposing Kronecker property, we obtain the following result: w j l n i l ds = (ax + b)n x + (ay + c)n y ds = e i a e i x2 x 1 (x 1 n x + y 1 n y ) 1 n y dx + a(x 1 n x + y 1 n y ) x 1 x 2 n y x2 x 1 (bn x + cn y ) 1 n y dx = + (bn x + cn y ) x 1 x 2 n y Since x 1 x 2 = is equal to n y (x2 x 1 ) 2 + (y 2 y 1 ) 2 = α 2 + β 2 we obtain: = δ ij e i w j l n i l ds = a(x 1 α + y 1 β) + (bα + cβ) = δ ij A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
20 The coefficients a, b, c are chosen by the 2D classical Galerkin functions, N i, i = 1, 2, 3, whose gradients are normal to the edges e i+1 for i = 1, 2 while the gradient of N 3 is normal to e 1. They can be written as: N i = a i + b i x + c i y, 2 T l where T l is the area of the triangle T l that can be written as: T l = 1 1 x 1 y x 2 y 2 1 x 3 y 3 A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
21 In particular, the coefficients are: a 1 = x 2 y 2 x 3 y 3 a 2 = x 1 y 1 x 3 y 3 a 3 = x 1 y 1 x 2 y 2 b 1 = 1 y 2 1 y 3 b 2 = 1 y 1 1 y 3 b 3 = 1 y 1 1 y 2 c 1 = 1 x 2 1 x 3 c 2 = 1 x 1 1 x 3 c 3 = 1 x 1 1 x 2 A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
22 It is simple to see that, for example, for the edge e 1 the following relations hold: αx 1 + βy 2 = a 3, α = b 3, β = c 3 Thus w j n 1 ds = a 3 a + (b 3 b + c 3 c). e 1 The unknowns are now the coefficients a, b, c of w j. Setting j = 1, from Kronecker property and substituting in the previous equation, we get: Note that 3 b i = i=1 a 3 a + b 3 b + c 3 c = 1 a 1 a + b 1 b + c 1 c = 0 a 2 a + b 2 b + c 2 c = 0 3 i=1 c i =0 and that 3 i=1 a i = 2 T l. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
23 The system gives the following relationship between a and T l : 2 T l a = 1 a = 1 2 T l. Application of divergence theorem to T w 1 dx dy assures that the value of a is the same along the three edges of T l and is positive and equal to 1 2 T l. Indeed w 1 dx dy = w 1 n ds = T T 3 i=1 e i w 1 n i ds = 1 But, T w 1 dx dy = T ( w 1 x + w 1 y ) ds = 2a ds = 2a T l T A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
24 Again, considering the first equation of the previous system and the measure of T l, we obtain for w 1 : b 1 l = x 3 2 T l, c1 l = y 3 2 T l. Similarly, the coefficients for w 2 and w 3 are: b j l = x j 1 2 T l, cj l = y j 1 2 T l, j = 2, 3 while a j l does not change, a j l = 1. Observe that with this 2 T l choice of basis for the V h space, the components g 1, g 2 and g 3 of G l on the element T l are the edge fluxes that is the velocity (or flux) along each edge of T l. A. Mazzia (DMMMSA) MHFE method a.a. 2007/ / 24
Finite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationMath 660-Lecture 15: Finite element spaces (I)
Math 660-Lecture 15: Finite element spaces (I) (Chapter 3, 4.2, 4.3) Before we introduce the concrete spaces, let s first of all introduce the following important lemma. Theorem 1. Let V h consists of
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods
More informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationBasic Principles of Weak Galerkin Finite Element Methods for PDEs
Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationDivergence-free or curl-free finite elements for solving the curl-div system
Divergence-free or curl-free finite elements for solving the curl-div system Alberto Valli Dipartimento di Matematica, Università di Trento, Italy Joint papers with: Ana Alonso Rodríguez Dipartimento di
More informationb i (x) u + c(x)u = f in Ω,
SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 1938 1953 c 2002 Society for Industrial and Applied Mathematics SUBOPTIMAL AND OPTIMAL CONVERGENCE IN MIXED FINITE ELEMENT METHODS ALAN DEMLOW Abstract. An elliptic
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationSolving the curl-div system using divergence-free or curl-free finite elements
Solving the curl-div system using divergence-free or curl-free finite elements Alberto Valli Dipartimento di Matematica, Università di Trento, Italy or: Why I say to my students that divergence-free finite
More informationMODEL OF GROUNDWATER FLOW IN FRACTURED ENVIRONMENT
Proceedings of ALGORITMY 2002 Conference on Scientific Computing, pp. 138 145 MODEL OF GROUNDWATER FLOW IN FRACTURED ENVIRONMENT JIŘÍ MARYŠKA, OTTO SEVERÝN AND MARTIN VOHRALÍK Abstract. A stochastic discrete
More informationA NUMERICAL APPROXIMATION OF NONFICKIAN FLOWS WITH MIXING LENGTH GROWTH IN POROUS MEDIA. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXX, 1(21, pp. 75 84 Proceedings of Algoritmy 2 75 A NUMERICAL APPROXIMATION OF NONFICIAN FLOWS WITH MIXING LENGTH GROWTH IN POROUS MEDIA R. E. EWING, Y. LIN and J. WANG
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationINTRODUCTION TO FINITE ELEMENT METHODS ON ELLIPTIC EQUATIONS LONG CHEN
INTROUCTION TO FINITE ELEMENT METHOS ON ELLIPTIC EQUATIONS LONG CHEN CONTENTS 1. Poisson Equation 1 2. Outline of Topics 3 2.1. Finite ifference Method 3 2.2. Finite Element Method 3 2.3. Finite Volume
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationScientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1
Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationA Stable Mixed Finite Element Scheme for the Second Order Elliptic Problems
A Stable Mixed Finite Element Scheme for the Second Order Elliptic Problems Miaochan Zhao, Hongbo Guan, Pei Yin Abstract A stable mixed finite element method (MFEM) for the second order elliptic problems,
More informationWEAK GALERKIN FINITE ELEMENT METHODS ON POLYTOPAL MESHES
INERNAIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 12, Number 1, Pages 31 53 c 2015 Institute for Scientific Computing and Information WEAK GALERKIN FINIE ELEMEN MEHODS ON POLYOPAL MESHES LIN
More informationMULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday.
MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* DOUGLAS N ARNOLD, RICHARD S FALK, and RAGNAR WINTHER Dedicated to Professor Jim Douglas, Jr on the occasion of his seventieth birthday Abstract
More information1. Introduction. The Stokes problem seeks unknown functions u and p satisfying
A DISCRETE DIVERGENCE FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU, JUNPING WANG, AND XIU YE Abstract. A discrete divergence free weak Galerkin finite element method is developed
More informationCoupling Non-Linear Stokes and Darcy Flow using Mortar Finite Elements
Coupling Non-Linear Stokes and Darcy Flow using Mortar Finite Elements V.J. Ervin E.W. Jenkins S. Sun Department of Mathematical Sciences Clemson University, Clemson, SC, 29634-0975, USA. Abstract We study
More informationAdaptive Finite Element Methods Lecture Notes Winter Term 2017/18. R. Verfürth. Fakultät für Mathematik, Ruhr-Universität Bochum
Adaptive Finite Element Methods Lecture Notes Winter Term 2017/18 R. Verfürth Fakultät für Mathematik, Ruhr-Universität Bochum Contents Chapter I. Introduction 7 I.1. Motivation 7 I.2. Sobolev and finite
More informationDecidability of consistency of function and derivative information for a triangle and a convex quadrilateral
Decidability of consistency of function and derivative information for a triangle and a convex quadrilateral Abbas Edalat Department of Computing Imperial College London Abstract Given a triangle in the
More informationFinite Elements. Colin Cotter. January 15, Colin Cotter FEM
Finite Elements January 15, 2018 Why Can solve PDEs on complicated domains. Have flexibility to increase order of accuracy and match the numerics to the physics. has an elegant mathematical formulation
More informationLECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,
LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical
More informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationChapter 3 Conforming Finite Element Methods 3.1 Foundations Ritz-Galerkin Method
Chapter 3 Conforming Finite Element Methods 3.1 Foundations 3.1.1 Ritz-Galerkin Method Let V be a Hilbert space, a(, ) : V V lr a bounded, V-elliptic bilinear form and l : V lr a bounded linear functional.
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
MATHEMATICS OF COMPUTATION Volume 75, Number 256, October 2006, Pages 1659 1674 S 0025-57180601872-2 Article electronically published on June 26, 2006 ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED
More informationMimetic Finite Difference methods
Mimetic Finite Difference methods An introduction Andrea Cangiani Università di Roma La Sapienza Seminario di Modellistica Differenziale Numerica 2 dicembre 2008 Andrea Cangiani (IAC CNR) mimetic finite
More informationOverlapping Schwarz preconditioners for Fekete spectral elements
Overlapping Schwarz preconditioners for Fekete spectral elements R. Pasquetti 1, L. F. Pavarino 2, F. Rapetti 1, and E. Zampieri 2 1 Laboratoire J.-A. Dieudonné, CNRS & Université de Nice et Sophia-Antipolis,
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationDiscontinuous Petrov-Galerkin Methods
Discontinuous Petrov-Galerkin Methods Friederike Hellwig 1st CENTRAL School on Analysis and Numerics for Partial Differential Equations, November 12, 2015 Motivation discontinuous Petrov-Galerkin (dpg)
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 3: Finite Elements in 2-D Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods 1 / 18 Outline 1 Boundary
More informationPDE & FEM TERMINOLOGY. BASIC PRINCIPLES OF FEM.
PDE & FEM TERMINOLOGY. BASIC PRINCIPLES OF FEM. Sergey Korotov Basque Center for Applied Mathematics / IKERBASQUE http://www.bcamath.org & http://www.ikerbasque.net 1 Introduction The analytical solution
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More informationRobust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms
www.oeaw.ac.at Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms Y. Efendiev, J. Galvis, R. Lazarov, J. Willems RICAM-Report 2011-05 www.ricam.oeaw.ac.at
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationTraces and Duality Lemma
Traces and Duality Lemma Recall the duality lemma with H / ( ) := γ 0 (H ()) defined as the trace space of H () endowed with minimal extension norm; i.e., for w H / ( ) L ( ), w H / ( ) = min{ ŵ H () ŵ
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationOptimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36
Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences,
More informationAn a posteriori error estimate and a Comparison Theorem for the nonconforming P 1 element
Calcolo manuscript No. (will be inserted by the editor) An a posteriori error estimate and a Comparison Theorem for the nonconforming P 1 element Dietrich Braess Faculty of Mathematics, Ruhr-University
More informationOverlapping Schwarz Preconditioners for Spectral. Problem in H(curl)
Overlapping Schwarz Preconditioners for Spectral Nédélec Elements for a Model Problem in H(curl) Technical Report TR2002-83 November 22, 2002 Department of Computer Science Courant Institute of Mathematical
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationH(div) and H(curl)-conforming Virtual Element Methods
Noname manuscript No. (will be inserted by the editor) H(div) and H(curl)-conforming Virtual Element Methods L. Beirão da Veiga F. Brezzi L. D. Marini A. Russo the date of receipt and acceptance should
More informationIntroduction. J.M. Burgers Center Graduate Course CFD I January Least-Squares Spectral Element Methods
Introduction In this workshop we will introduce you to the least-squares spectral element method. As you can see from the lecture notes, this method is a combination of the weak formulation derived from
More informationA WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS
A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS LIN MU, JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. This article introduces and analyzes a weak Galerkin mixed finite element method
More informationNon-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions
Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions Wayne McGee and Padmanabhan Seshaiyer Texas Tech University, Mathematics and Statistics (padhu@math.ttu.edu) Summary. In
More informationWEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER PARABOLIC EQUATIONS
INERNAIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 13, Number 4, Pages 525 544 c 216 Institute for Scientific Computing and Information WEAK GALERKIN FINIE ELEMEN MEHOD FOR SECOND ORDER PARABOLIC
More informationA multipoint flux mixed finite element method on hexahedra
A multipoint flux mixed finite element method on hexahedra Ross Ingram Mary F. Wheeler Ivan Yotov Abstract We develop a mixed finite element method for elliptic problems on hexahedral grids that reduces
More informationNew class of finite element methods: weak Galerkin methods
New class of finite element methods: weak Galerkin methods Xiu Ye University of Arkansas at Little Rock Second order elliptic equation Consider second order elliptic problem: a u = f, in Ω (1) u = 0, on
More informationQUADRILATERAL H(DIV) FINITE ELEMENTS
QUADRILATERAL H(DIV) FINITE ELEMENTS DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK Abstract. We consider the approximation properties of quadrilateral finite element spaces of vector fields defined
More informationA DELTA-REGULARIZATION FINITE ELEMENT METHOD FOR A DOUBLE CURL PROBLEM WITH DIVERGENCE-FREE CONSTRAINT
A DELTA-REGULARIZATION FINITE ELEMENT METHOD FOR A DOUBLE CURL PROBLEM WITH DIVERGENCE-FREE CONSTRAINT HUOYUAN DUAN, SHA LI, ROGER C. E. TAN, AND WEIYING ZHENG Abstract. To deal with the divergence-free
More informationProjection Theorem 1
Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality
More informationA Locking-Free MHM Method for Elasticity
Trabalho apresentado no CNMAC, Gramado - RS, 2016. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics A Locking-Free MHM Method for Elasticity Weslley S. Pereira 1 Frédéric
More informationSolutions of Selected Problems
1 Solutions of Selected Problems October 16, 2015 Chapter I 1.9 Consider the potential equation in the disk := {(x, y) R 2 ; x 2 +y 2 < 1}, with the boundary condition u(x) = g(x) r for x on the derivative
More informationPARTITION OF UNITY FOR THE STOKES PROBLEM ON NONMATCHING GRIDS
PARTITION OF UNITY FOR THE STOES PROBLEM ON NONMATCHING GRIDS CONSTANTIN BACUTA AND JINCHAO XU Abstract. We consider the Stokes Problem on a plane polygonal domain Ω R 2. We propose a finite element method
More informationLocal discontinuous Galerkin methods for elliptic problems
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2002; 18:69 75 [Version: 2000/03/22 v1.0] Local discontinuous Galerkin methods for elliptic problems P. Castillo 1 B. Cockburn
More informationA FAMILY OF MULTISCALE HYBRID-MIXED FINITE ELEMENT METHODS FOR THE DARCY EQUATION WITH ROUGH COEFFICIENTS. 1. Introduction
A FAMILY OF MULTISCALE HYBRID-MIXED FINITE ELEMENT METHODS FOR THE DARCY EQUATION WITH ROUGH COEFFICIENTS CHRISTOPHER HARDER, DIEGO PAREDES 2, AND FRÉDÉRIC VALENTIN Abstract. We aim at proposing novel
More informationYongdeok Kim and Seki Kim
J. Korean Math. Soc. 39 (00), No. 3, pp. 363 376 STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM Yongdeok Kim and Seki Kim Abstract. Stability result is obtained for
More informationarxiv: v2 [math.na] 23 Apr 2016
Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements arxiv:508.009v2 [math.na] 23 Apr 206 Zhiqiang Cai Cuiyu He Shun Zhang May 2, 208 Abstract. In [8], we introduced
More informationAn Extended Finite Element Method for a Two-Phase Stokes problem
XFEM project An Extended Finite Element Method for a Two-Phase Stokes problem P. Lederer, C. Pfeiler, C. Wintersteiger Advisor: Dr. C. Lehrenfeld August 5, 2015 Contents 1 Problem description 2 1.1 Physics.........................................
More informationWeak Galerkin Finite Element Scheme and Its Applications
Weak Galerkin Finite Element Scheme and Its Applications Ran Zhang Department of Mathematics Jilin University, China IMS, Singapore February 6, 2015 Talk Outline Motivation WG FEMs: Weak Operators + Stabilizer
More informationA note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations
A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations Bernardo Cockburn Guido anschat Dominik Schötzau June 1, 2007 Journal of Scientific Computing, Vol. 31, 2007, pp.
More informationLocal flux mimetic finite difference methods
Local flux mimetic finite difference methods Konstantin Lipnikov Mikhail Shashkov Ivan Yotov November 4, 2005 Abstract We develop a local flux mimetic finite difference method for second order elliptic
More informationFundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural
More information4 Divergence theorem and its consequences
Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........
More informationSolutions 2017 AB Exam
1. Solve for x : x 2 = 4 x. Solutions 2017 AB Exam Texas A&M High School Math Contest October 21, 2017 ANSWER: x = 3 Solution: x 2 = 4 x x 2 = 16 8x + x 2 x 2 9x + 18 = 0 (x 6)(x 3) = 0 x = 6, 3 but x
More informationA Multiscale Mortar Multipoint Flux Mixed Finite Element Method
A Multiscale Mortar Multipoint Flux Mixed Finite Element Method Mary F. Wheeler Guangri Xue Ivan Yotov Abstract In this paper, we develop a multiscale mortar multipoint flux mixed finite element method
More informationA POSTERIORI ERROR ESTIMATES OF TRIANGULAR MIXED FINITE ELEMENT METHODS FOR QUADRATIC CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEMS
A POSTERIORI ERROR ESTIMATES OF TRIANGULAR MIXED FINITE ELEMENT METHODS FOR QUADRATIC CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEMS Z. LU Communicated by Gabriela Marinoschi In this paper, we discuss a
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationFINITE VOLUME METHOD: BASIC PRINCIPLES AND EXAMPLES
FINITE VOLUME METHOD: BASIC PRINCIPLES AND EXAMPLES SHRUTI JAIN B.Tech III Year, Electronics and Communication IIT Roorkee Tutors: Professor G. Biswas Professor S. Chakraborty ACKNOWLEDGMENTS I would like
More informationApplied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.
Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define
More informationON LEAST-SQUARES FINITE ELEMENT METHODS FOR THE POISSON EQUATION AND THEIR CONNECTION TO THE DIRICHLET AND KELVIN PRINCIPLES
ON LEAST-SQUARES FINITE ELEMENT METHODS FOR THE POISSON EQUATION AND THEIR CONNECTION TO THE DIRICHLET AND KELVIN PRINCIPLES PAVEL BOCHEV 1,3 AND MAX GUNZBURGER 2 Abstract. Least-squares finite element
More informationGAKUTO International Series
GAKUTO International Series Mathematical Sciences and Applications, Vol.28(2008) Proceedings of Fourth JSIAM-SIMMAI Seminar on Industrial and Applied Mathematics, pp.139-148 A COMPUTATIONAL APPROACH TO
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationThe Mortar Boundary Element Method
The Mortar Boundary Element Method A Thesis submitted for the degree of Doctor of Philosophy by Martin Healey School of Information Systems, Computing and Mathematics Brunel University March 2010 Abstract
More informationHIGH DEGREE IMMERSED FINITE ELEMENT SPACES BY A LEAST SQUARES METHOD
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 14, Number 4-5, Pages 604 626 c 2017 Institute for Scientific Computing and Information HIGH DEGREE IMMERSED FINITE ELEMENT SPACES BY A LEAST
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More informationFast Marching Methods for Front Propagation Lecture 1/3
Outline for Front Propagation Lecture 1/3 M. Falcone Dipartimento di Matematica SAPIENZA, Università di Roma Autumn School Introduction to Numerical Methods for Moving Boundaries ENSTA, Paris, November,
More informationChapter 12. Partial di erential equations Di erential operators in R n. The gradient and Jacobian. Divergence and rotation
Chapter 12 Partial di erential equations 12.1 Di erential operators in R n The gradient and Jacobian We recall the definition of the gradient of a scalar function f : R n! R, as @f grad f = rf =,..., @f
More informationIntroduction to finite element exterior calculus
Introduction to finite element exterior calculus Ragnar Winther CMA, University of Oslo Norway Why finite element exterior calculus? Recall the de Rham complex on the form: R H 1 (Ω) grad H(curl, Ω) curl
More informationAn Adaptive Mixed Finite Element Method using the Lagrange Multiplier Technique
An Adaptive Mixed Finite Element Method using the Lagrange Multiplier Technique by Michael Gagnon A Project Report Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment
More informationOn atomistic-to-continuum couplings without ghost forces
On atomistic-to-continuum couplings without ghost forces Dimitrios Mitsoudis ACMAC Archimedes Center for Modeling, Analysis & Computation Department of Applied Mathematics, University of Crete & Institute
More informationMixed Finite Elements Method
Mixed Finite Elements Method A. Ratnani 34, E. Sonnendrücker 34 3 Max-Planck Institut für Plasmaphysik, Garching, Germany 4 Technische Universität München, Garching, Germany Contents Introduction 2. Notations.....................................
More informationA Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions
A Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions Zhiqiang Cai Seokchan Kim Sangdong Kim Sooryun Kong Abstract In [7], we proposed a new finite element method
More informationHigh order, finite volume method, flux conservation, finite element method
FLUX-CONSERVING FINITE ELEMENT METHODS SHANGYOU ZHANG, ZHIMIN ZHANG, AND QINGSONG ZOU Abstract. We analyze the flux conservation property of the finite element method. It is shown that the finite element
More informationAn interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes
An interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes Vincent Heuveline Friedhelm Schieweck Abstract We propose a Scott-Zhang type interpolation
More informationGlowinski Pironneau method for the 3D ω-ψ equations
280 GUERMOND AND QUARTAPELLE Glowinski Pironneau method for the 3D ω-ψ equations Jean-Luc Guermond and Luigi Quartapelle 1 LIMSI CNRS, Orsay, France, and Dipartimento di Fisica, Politecnico di Milano,
More informationDispersive nonlinear partial differential equations
Dispersive nonlinear partial differential equations Elena Kartashova 16.02.2007 Elena Kartashova (RISC) Dispersive nonlinear PDEs 16.02.2007 1 / 25 Mathematical Classification of PDEs based on the FORM
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationHybridized Discontinuous Galerkin Methods
Hybridized Discontinuous Galerkin Methods Theory and Christian Waluga Joint work with Herbert Egger (Uni Graz) 1st DUNE User Meeting, Stuttgart Christian Waluga (AICES) HDG Methods October 6-8, 2010 1
More informationProjected Surface Finite Elements for Elliptic Equations
Available at http://pvamu.edu/aam Appl. Appl. Math. IN: 1932-9466 Vol. 8, Issue 1 (June 2013), pp. 16 33 Applications and Applied Mathematics: An International Journal (AAM) Projected urface Finite Elements
More information