Texts in Computational Science and Engineering 10

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1 Texts in Computational Science and Engineering 10 Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick For further volumes:

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3 Mats G. Larson Fredrik Bengzon The Finite Element Method: Theory, Implementation, and Applications 123

4 Mats G. Larson Fredrik Bengzon Department of Mathematics Umeå University Umeå Sweden ISSN ISBN ISBN (ebook) DOI / Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Mathematics Subject Classification (2010): 65N30, 65N15, 65M60, 65M15 c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB 1 and its PDE-Toolbox. A basic knowledge of the MATLAB language is therefore necessary. The PDE-Toolbox is used for pre and post processing (i.e., meshing and plotting). We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics. The book is loosely divided into two parts Chaps. 1 6 which provides basic material and Chaps which covers more advanced material and applications. In the first part Chaps. 1 4 gives an introduction to the finite element method for stationary second order elliptic problems. Here we consider the one dimensional case in Chaps. 1 and 2 and then extend to two dimensions in Chaps. 3 and 4. In Chap. 5 we consider time dependent problems and in Chap. 6 we give an introduction to numerical linear algebra for sparse linear systems of equations. In the second more advanced part we present the abstract theory in Chap. 7, various finite elements in Chap. 8, and a short introduction to nonlinear problems 1 MATLAB is a registered trademark of The MathWorks Inc. ( v

6 vi Preface in Chap. 9. In Chaps we consider applications to transport problems, solid mechanics, fluid mechanics, and electromagnetics. Finally, in Chap. 14 we give a short introduction to discontinuous Galerkin methods. The book is based on lecture notes used in various courses given by the authors and their coworkers during the last eight years at Chalmers University of Technology, Umeå University, Uppsala University, and at the University of Oslo. Several different courses in engineering and mathematics programs can be based on the material in the book. The following are some examples of possible courses that we have had in mind while developing the material: Short introduction to finite elements as part of a calculus or numerical analysis course. Chapters 1 and 2. Introduction to finite elements as part of a calculus or numerical analysis course, with focus on one dimensional stationary and time dependent problems. Chapters 1 and 2 and parts of Chapter 5. Introduction to finite elements as part of a calculus or numerical analysis course, with focus on stationary problems in one and two dimensions. Chapters 1 4. First course on finite elements. Chapters 1 6. Second course on finite elements and its applications. Chapters Umeå, Sweden Mats G. Larson Fredrik Bengzon

7 Acknowledgements This book is based on material developed for courses given at Chalmers University of Technology, Umeå University, Uppsala University, and the University of Oslo during the last 8 years and the authors gratefully acknowledge the contributions of the teachers and students involved. We also wish to thank Timothy J. Barth, (Nasa Ames), Axel Målqvist (Uppsala University), Hans-Petter Langtangen (Simula Research Laboratory and Oslo University), Thomas Ericsson (Chalmers University of Technology), and Håkan Jakobsson, Robert Söderlund, Karl Larsson, Per Vesterlund, and Jakob Öhrman (Umeå University), for valuable discussions and comments on the manuscript. vii

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9 Contents 1 Piecewise Polynomial Approximation in 1D Piecewise Polynomial Spaces The Space of Linear Polynomials The Space of Continuous Piecewise Linear Polynomials Interpolation Linear Interpolation Continuous Piecewise Linear Interpolation L 2 -Projection Definition Derivation of a Linear System of Equations Basic Algorithm to Compute the L 2 -Projection A Priori Error Estimate Quadrature The Mid-Point Rule The Trapezoidal Rule Simpson s Formula Computer Implementation Assembly of the Mass Matrix Assembly of the Load Vector Problems The Finite Element Method in 1D The Finite Element Method for a Model Problem A Two-point Boundary Value Problem Variational Formulation Finite Element Approximation Derivation of a Linear System of Equations Basic Algorithm to Compute the Finite Element Solution A Priori Error Estimate ix

10 x Contents 2.2 Mathematical Modeling Derivation of the Stationary Heat Equation Boundary Conditions for the Heat Equation Dirichlet Boundary Conditions Neumann Boundary Conditions Robin Conditions Derivation of a Differential Equation for the Deformation of a Bar Boundary Conditions for the Bar Dirichlet Boundary Conditions Neumann Boundary Conditions Robin Boundary Conditions A Model Problem with Variable Coefficients Variational Formulation Finite Element Approximation Computer Implementation Assembly of the Stiffness Matrix and Load Vector A Finite Element Solver for a General Two-Point Boundary Value Problem Adaptive Finite Element Methods A Posteriori Error Estimate Adaptive Mesh Refinement Further Reading Problems Piecewise Polynomial Approximation in 2D Meshes Triangulations Data Structures for a Triangulation Mesh Generation Piecewise Polynomial Spaces The Space of Linear Polynomials The Space of Continuous Piecewise Linear Polynomials The Space of Piecewise Constants Interpolation Linear Interpolation Continuous Piecewise Linear Interpolation L 2 -Projection Definition Derivation of a Linear System of Equations Basic Algorithm to Compute the L 2 -Projection Existence and Uniqueness of the L 2 -Projection A Priori Error Estimate Quadrature and Numerical Integration... 59

11 Contents xi 3.6 Computer Implementation Assembly of the Mass Matrix Assembly of the Load Vector Properties of the Mass Matrix Further Reading Problems The Finite Element Method in 2D Green s Formula The Finite Element Method for Poisson s Equation Poisson s Equation Variational Formulation Finite Element Approximation Derivation of a Linear System of Equations Basic Algorithm to Compute the Finite Element Solution Some Useful Inequalities Basic Analysis of the Finite Element Method Existence and Uniqueness of the Finite Element Solution A Priori Error Estimates Properties of the Stiffness Matrix A Model Problem with Variable Coefficients Computer Implementation Assembly of the Stiffness Matrix Assembling the Boundary Conditions A Finite Element Solver for Poisson s Equation Potential Flow Over a Wing The Dirichlet Problem The Neumann Problem The Eigenvalue Problem Adaptive Finite Element Methods A Posteriori Error Estimate Adaptive Mesh Refinement Rivara Refinement Regular Refinement Adaptive Finite Elements Using MATLAB Duality Based a Posteriori Error Estimate Further Reading Problems Time-Dependent Problems Finite Difference Methods for Systems of ODE The Heat Equation Derivation of the Transient Heat Equation Model Problem

12 xii Contents Variational Formulation Spatial Discretization Time Discretization Stability Estimates A Space Discrete Estimate A Fully Discrete Estimate A Priori Error Estimates Ritz Projection A Space Discrete Estimate A Fully Discrete Estimate Computer Implementation Mass Lumping Space-Time Finite Element Approximation A Posteriori Error Estimate The Wave Equation Derivation of the Acoustic Wave Equation Model Problem Variational Formulation Spatial Discretization Time Discretization Conservation of Energy Computer Implementation Further Reading Problems Solving Large Sparse Linear Systems Linear Systems Direct Methods Triangular Systems, Backward and Forward Substitution Fill-ins, Graphs, and Symbolic and Numeric Factorization Matrix Reorderings Minimum Degree Nested Dissection Iterative Methods Basic Iterative Methods The Jacobi Method The Gauss-Seidel Method Convergence Analysis Projection Methods One-Dimensional Projection Methods Krylov Subspaces The Conjugate Gradient Method The Generalized Minimal Residual Method Other Krylov Methods

13 Contents xiii 6.4 Preconditioning Jacobi Preconditioning Polynomial Preconditioners Incomplete Factorizations Towards Multigrid The Smoothing Property A Two-Grid Method Further Reading Problems Abstract Finite Element Analysis Function Spaces Normed and Inner Product Vector Spaces Banach and Hilbert Spaces L p Spaces Weak Derivatives Sobolev Spaces Traces Interpolation of Functions in Hilbert Spaces The Clement Interpolant The Scott-Zhang Interpolant The Abstract Setting An Abstract Variational Problem The Riesz Representation Theorem An Equivalent Minimization Problem The Lax-Milgram Lemma Application to Elliptic Partial Differential Equations Poisson s Equation The Diffusion-Reaction Equation Laplace Equation with Mixed Boundary Conditions A General Linear Second Order Elliptic Problem Abstract Finite Element Approximation Abstract Finite Element Method The Stiffness Matrix and Load Vector Galerkin Orthogonality A Priori Error Estimates A Posteriori Error Estimates Further Reading Problems The Finite Element Different Types of Finite Elements Formal Definition of a Finite Element Shape Functions for the Linear Lagrange Triangle Shape Functions for the Quadratic Lagrange Triangle

14 xiv Contents Higher Order Triangular Lagrange Elements Shape Functions for the Bilinear Quadrilateral Element Shape Functions for the Linear Lagrange Tetrahedron Shape Functions for the Trilinear Brick Element The Isoparametric Mapping Numerical Quadrature Revisited Renumbering the Mesh for Quadratic Nodes Assembly of the Isoparametric Quadratic Stiffness Matrix Some Exotic Finite Elements The Morley Element The Crouzeix-Raviart Element The Lowest Order Raviart-Thomas Element The Lowest Order Nédélec Element Further Reading Problems Non-linear Problems Piccard Iteration Newton s Method The Non-linear Poisson Equation The Newton-Galerkin Method Finite Element Approximation Computer Implementation Piccard Iteration as a Simplified Newton Method The Bistable Equation Weak Form Space Discretization Time Discretization Piccard Iteration Newton s Method Numerical Approximations of the Jacobian Difference Quotient Approximation Broyden s Method Further Reading Problems Transport Problems The Transport Equation Weak Form Standard Galerkin Finite Element Approximation Computer Implementation The Need for Stabilization Least-Squares Stabilization

15 Contents xv GLS Finite Element Approximation A Priori Error Estimate Computer Implementation Heat Transfer in a Fluid Flow Further Reading Problems Solid Mechanics Governing Equations Cauchy s Equilibrium Equation Constitutive Equations and Hooke s Law Boundary Conditions Linear Elastostatics Weak Form Existence and Uniqueness of the Solution Finite Element Approximation A Priori Error Estimate Engineering Notation Computer Implementation Verifying the Energy Norm Convergence A Posteriori Error Estimate Adaptive Mesh Refinement on a Rotated L-Shaped Domain Linear Thermoelasticity Linear Elastodynamics Modal Analysis Eigenvalues and Eigenmodes of a Steel Bracket Time Stepping Rayleigh Damping Newmark s Method Thin Plates Governing Equations Weak Form The C 1 Continuity Requirement and the Morley Approximation Further Reading Problems Fluid Mechanics Governing Equations Conservation of Mass Momentum Balance Incompressible Newtonian Fluids Boundary- and Initial Conditions

16 xvi Contents 12.2 The Stokes System The Stationary Stokes System Weak Form Equivalent Optimization Problem The Continuous Inf-Sup Condition Finite Element Approximation The Discrete Inf-Sup Condition The Uzawa Algorithm Three Inf-Sup Stable Finite Elements The Taylor-Hood Element The MINI Element The Non-conforming P 1 P 0 Element Asserting the Inf-Sup Condition Fortin s Trick A Priori Error Estimate Computer Implementation The Lid-Driven Cavity A Posteriori Error Estimate Stabilized Finite Element Methods Spurious Pressure Modes Galerkin Least Squares Stabilization Macroelement Techniques The Navier-Stokes Equations The Fractional Step Method Time Discretization Space Discretization Computer Implementation The DFG Benchmark Further Reading Problems Electromagnetics Governing Equations Constitutive Equations Gauss Laws Ampère s Law Faraday s Law Maxwell s Equations Initial- and Boundary Conditions The Electric and Magnetic Wave Equations The Time Harmonic Assumption Electro- and Magnetostatics The Time Harmonic Electric Wave Equation Boundary Conditions Green s Formula Revisited Weak Form

17 Contents xvii Finite Element Approximation Computer Implementation Electromagnetic Scattering from a Cylinder The Magnetostatic Potential Equation Boundary Conditions Weak Form The Helmholtz Decomposition Existence and Uniqueness of the Solution Finite Element Approximation A Priori Error Estimate A Posteriori Error Estimate Further Reading Problems Discontinuous Galerkin Methods A First Order Problem Model Problem Discontinuous Finite Element Spaces The Discontinuous Galerkin Method Stability Estimates A Priori Error Estimates A Second Order Elliptic Problem Model Problem The Symmetric Interior Penalty Method Approximation Properties A Priori Error Estimates Non-symmetric Versions of the Method Computer Implementation The Transport Equation with Diffusion Further Reading Problems A Some Additional Matlab Code A.1 Tri2Edge.m A.2 EdgeMidPoints.m A.3 Tri2Tri.m A.4 Dslitg.m A.5 Airfoilg.m A.6 RectCircg.m A.7 DFGg.m A.8 Scatterg.m References Index