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Partial differential equations and the finite element method
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Partial Differential Equations
and the Finite Element Method
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN 111, DAVID A. COX, PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
Partial Differential Equations
and the Finite Element Method
Pave1 Solin
The University of Texas at El Paso
Academy of Sciences ofthe Czech Republic
@ZEicIENCE
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 0 2006 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cutuloging-in-Publication Lhta:
Solin. Pavel.
p. cm.
Partial differential equations and the finite element method I Pave1 Solin
Includes bibliographical references and index.
ISBN-I 3 978-0-47 1-72070-6
ISBN-I0 0-471-72070-4 (cloth : acid-free paper)
Title.
I. Differential equations, Partial-Numerical solutions. 2. Finite clement method. 1.
QA377.S65 2005
5 18'.64-dc22 200548622
Printed in the United States of America
I0987654321
To Dagmar
CONTENTS
List of Figures
List of Tables
Preface
Acknowledgments
1 Partial Differential Equations
1.1 Selected general properties
1.1.1 Classification and examples
1.1.2 Hadamard’s well-posedness
1.1.3
1.1.4 Exercises
General existence and uniqueness results
1.2 Second-order elliptic problems
1.2.1 Weak formulation of a model problem
1.2.2 Bilinear forms, energy norm, and energetic inner product
1.2.3 The Lax-Milgram lemma
1.2.4 Unique solvability of the model problem
1.2.5 Nonhomogeneous Dirichlet boundary conditions
1.2.6 Neumann boundary conditions
1.2.7 Newton (Robin) boundary conditions
1.2.8 Combining essential and natural boundary conditions
xv
xxi
xxiii
xxv
1
2
2
5
9
11
13
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16
18
18
19
21
22
23
vii
Viii CONTENTS
1.2.9 Energy of elliptic problems
1.2.10 Maximum principles and well-posedness
1.2.1 1 Exercises
1.3 Second-order parabolic problems
1.3.1 Initial and boundary conditions
1.3.2 Weak formulation
I .3.3
1.3.4 Exercises
Existence and uniqueness of solution
1.4 Second-order hyperbolic problems
1.4.1 Initial and boundary conditions
1.4.2
1.4.3 The wave equation
I .4.4 Exercises
Weak formulation and unique solvability
1 .5 First-order hyperbolic problems
1.5.1 Conservation laws
1 S.2 Characteristics
1 S.3
1 S.4 Riemann problem
1 S.5
1 S.6 Exercises
Exact solution to linear first-order systems
Nonlinear flux and shock formation
2 Continuous Elements for 1 D Problems
2.1 The general framework
2. I. 1
2.1.2
2.1.3
2. I .4
2. I .5 Exercises
The Galerkin method
Orthogonality of error and CCa’s lemma
Convergence of the Cialerkin method
Ritz method for symmetric problems
2.2 Lowest-order elements
2.2.1 Model problem
2.2.2
2.2.3 Piecewise-affine basis functions
2.2.4
2.2.5 Element-by-element assembling procedure
2.2.6 Refinement and convergence
2.2.7 Exercises
Finite-dimensional subspace V,, C v
The system of linear algebraic equations
2.3 Higher-order numerical quadrature
2.3.1 Gaussian quadrature rules
2.3.2 Selected quadrature constants
2.3.3 Adaptive quadrature
2.3.4 Exercises
2.4 Higher-order elements
24
26
29
30
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31
32
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33
34
34
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61
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66
CONTENTS ix
2.4.1
2.4.2
2.4.3
2.4.4
2.4.5
2.4.6
2.4.7
2.4.8
2.4.9
2.4.10
Motivation problem
Affine concept: reference domain and reference maps
Transformation of weak forms to the reference domain
Higher-order Lagrange nodal shape functions
Chebyshev and Gauss-Lobatto nodal points
Higher-order Lobatto hierarchic shape functions
Constructing basis of the space Vh,p
Data structures
Assembling algorithm
Exercises
2.5 The sparse stiffness matrix
2.5.1
2.5.2 Condition number
2.5.3 Conditioning of shape functions
2.5.4
2.5.5 Exercises
Compressed sparse row (CSR) data format
Stiffness matrix for the Lobatto shape functions
2.6 Implementing nonhomogeneous boundary conditions
2.6.1 Dirichlet boundary conditions
2.6.2
2.6.3 Exercises
Combination of essential and natural conditions
2.7 Interpolation on finite elements
2.7.1 The Hilbert space setting
2.7.2 Best interpolant
2.7.3 Projection-based interpolant
2.7.4 Nodal interpolant
2.7.5 Exercises
3 General Concept of Nodal Elements
3.1 The nodal finite element
3.1.1 Unisolvency and nodal basis
3.1.2 Checking unisolvency
Example: lowest-order Q' - and PI-elements
3.2.1 Q1-element
3.2.2 P1-element
3.2.3
3.2
Invertibility of the quadrilateral reference map z~
3.3 Interpolation on nodal elements
3.3.1 Local nodal interpolant
3.3.2 Global interpolant and conformity
3.3.3 Conformity to the Sobolev space H'
3.4 Equivalence of nodal elements
3.5 Exercises
66
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74
76
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82
84
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103
1 04
106
107
108
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113
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116
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122
X CONTENTS
4 Continuous Elements for 2D Problems
4.1 Lowest-order elements
4.1.1
4.1.2 Approximations and variational crimes
4.1.3
4.1.4
4.1.5
4.1.6 Connectivity arrays
4.1.7 Assembling algorithm for Q'/P'-elements
4.1.8 Lagrange interpolation on Q'/P'-meshes
4.1.9 Exercises
Higher-order numerical quadrature in 2D
4.2.1 Gaussian quadrature on quads
4.2.2 Gaussian quadrature on triangles
4.3.1 Product Gauss-Lobatto points
4.3.2 Lagrange-Gauss-Lobatto Qp,'-elements
4.3.3
4.3.4 The Fekete points
4.3.5 Lagrange-Fekete PP-elements
4.3.6
4.3.7 Data structures
4.3.8 Connectivity arrays
4.3.9 Assembling algorithm for QPIPp-elements
4.3.10 Lagrange interpolation on Qp/Pp-meshes
4.3.1 1 Exercises
Model problem and its weak formulation
Basis of the space Vh,p
Transformation of weak forms to the reference domain
Simplified evaluation of stiffness integrals
4.2
4.3 Higher-order nodal elements
Lagrange interpolation and the Lebesgue constant
Basis of the space v7,Tl
125
126
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166
5 Transient Problems and ODE Solvers
5.1 Method of lines
5.1.1 Model problem
5.1.2 Weak formulation
5.1.3 The ODE system
5.1.4
5.1.5
Construction of the initial vector
Autonomous systems and phase flow
One-step methods, consistency and convergence
Explicit and implicit Euler methods
5.2 Selected time integration schemes
5.2.1
5.2.2
5.2.3 Stiffness
5.2.4 Explicit higher-order RK schemes
5.2.5
5.2.6 General (implicit) RK schemes
Embedded RK methods and adaptivity
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168
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184
5.3 Introduction to stability
5.3.1 Autonomization of RK methods
5.3.2
5.3.3
5.3.4
5.3.5
5.3.6 A-stability and L-stability
5.4.1 Collocation methods
5.4.2
5.4.3 Solution of nonlinear systems
Stability of linear autonomous systems
Stability functions and stability domains
Stability functions for general RK methods
Maximum consistency order of IRK methods
5.4 Higher-order IRK methods
Gauss and Radau IRK methods
5.5 Exercises
6 Beam and Plate Bending Problems
6.1 Bending of elastic beams
6.1. I Euler-Bernoulli model
6.1.2 Boundary conditions
6.1.3 Weak formulation
6.1.4
Lowest-order Hermite elements in 1D
6.2.1 Model problem
6.2.2 Cubic Hermite elements
Higher-order Hermite elements in 1D
6.3.1 Nodal higher-order elements
6.3.2 Hierarchic higher-order elements
6.3.3 Conditioning of shape functions
6.3.4 Basis of the space Vh,p
6.3.5 Transformation of weak forms to the reference domain
6.3.6 Connectivity arrays
6.3.7 Assembling algorithm
6.3.8 Interpolation on Hermite elements
6.4.1 Lowest-order elements
6.4.2 Higher-order Hermite-Fekete elements
6.4.3 Design of basis functions
6.4.4
6.5.1 Reissner-Mindlin (thick) plate model
6.5.2 Kirchhoff (thin) plate model
6.5.3 Boundary conditions
6.5.4
6.5.5
Existence and uniqueness of solution
6.2
6.3
6.4 Hermite elements in 2D
Global nodal interpolant and conformity
6.5 Bending of elastic plates
Weak formulation and unique solvability
BabuSka’s paradox of thin plates
CONTENTS xi
185
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188
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200
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209
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xii CONTENTS
6.6 Discretization by H2-conforming elements
6.6.1
6.6.2 Local interpolant, conformity
6.6.3
6.6.4 Transformation to reference domains
6.6.5 Design of basis functions
6.6.6 Higher-order nodal Argyris-Fekete elements
Lowest-order (quintic) Argyris element, unisolvency
Nodal shape functions on the reference domain
6.7 Exercises
7 Equations of Electrornagnetics
7.1 Electromagnetic field and its basic characteristics
7.1.1 Integration along smooth curves
7.1.2
7.1.3
7.1.4
7.1 .5
7.1.6 Conductors and dielectrics
7.1.7 Magnetic materials
7.1.8 Conditions on interfaces
7.2.1 Scalar electric potential
7.2.2 Scalar magnetic potential
7.2.3
7.2.4
7.2.5 Other wave equations
Equations for the field vectors
7.3.1
7.3.2
7.3.3 Interface and boundary conditions
7.3.4 Time-harmonic Maxwell’s equations
7.3.5 Helmholtz equation
7.4.1 Normalization
7.4.2 Model problem
7.4.3 Weak formulation
7.4.4
Maxwell’s equations in integral form
Maxwell’s equations in differential form
Constitutive relations and the equation of continuity
Media and their characteristics
7.2 Potentials
Vector potential and gauge transformations
Potential formulation of Maxwell’s equations
7.3
Equation for the electric field
Equation for the magnetic field
7.4 Time-harmonic Maxwell’s equations
Existence and uniqueness of solution
7.5.1 Conformity requirements of the space H(cur1)
7.5.2 Lowest-order (Whitney) edge elements
7.5.3 Higher-order edge elements of NCdClec
7.5.4 Transformation of weak forms to the reference domain
7.5.5 Interpolation on edge elements
7.5 Edge elements
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28 1
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316
CONTENTS xiii
7.5.6
7.6 Exercises
Conformity of edge elements to the space H(cur1)
Appendix A: Basics of Functional Analysis
A. 1 Linear spaces
A. 1.1
A. 1.2
A. 1.3
A. 1.4
A. 1.5
A. 1.6
A. 1.7
A. 1.8
A. 1.9
A. 1.10
A. 1.1 1
A. 1.12 Exercises
Real and complex linear space
Checking whether a set is a linear space
Intersection and union of subspaces
Linear combination and linear span
Sum and direct sum of subspaces
Linear independence, basis, and dimension
Linear operator, null space, range
Composed operators and change of basis
Determinants, eigenvalues, and eigenvectors
Hermitian, symmetric, and diagonalizable matrices
Linear forms, dual space, and dual basis
A.2 Normed spaces
A.2.1 Norm and seminorm
A.2.2 Convergence and limit
A.2.3 Open and closed sets
A.2.4 Continuity of operators
A.2.5
A.2.6 Equivalence of norms
A.2.7 Banach spaces
A.2.8 Banach fixed point theorem
A.2.9 Lebesgue integral and LP-spaces
A.2.10 Basic inequalities in LP-spaces
A.2.11
A.2.12 Exercises
Operator norm and C(U, V) as a normed space
Density of smooth functions in LP-spaces
A.3 Inner product spaces
A.3.1 Inner product
A.3.2 Hilbert spaces
A.3.3 Generalized angle and orthogonality
A.3.4 Generalized Fourier series
A.3.5 Projections and orthogonal projections
A.3.6 Representation of linear forms (Riesz)
A.3.7 Compactness, compact operators, and the Fredholm alternative
A.3.8 Weak convergence
A.3.9 Exercises
A.4 Sobolev spaces
A.4.1 Domain boundary and its regularity
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32 1
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326
327
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339
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366
37 1
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380
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412
xiv CONTENTS
A.4.2
A.4.3
A.4.4
A.4.5
A.4.6
A.4.7
A.4.8
A.4.9
Distributions and weak derivatives
Spaces Wklp and Hk
Discontinuity of HI-functions in R", d 2 2
PoincarC-Friedrichs' inequality
Embeddings of Sobolev spaces
Traces of W"p-functions
Generalized integration by parts formulae
Exercises
Appendix B: Software and Examples
B. 1 Sparse Matrix Solvers
B. 1.1 The sMatrix utility
B. 1.2 An example application
B. 1.3 Interfacing with PETSc
B. 1.4 Interfacing with Trilinos
B. 1.5 Interfacing with UMFPACK
The High-Performance Modular Finite Element System HERMES
B.2.1 Modular structure of HERMES
B.2.2 The elliptic module
B.2.3 The Maxwell's module
B.2.4
B.2.5 Example 2: Insulator problem
B.2.6 Example 3: Sphere-cone problem
B.2.7
B.2.8 Example 5: Diffraction problem
B.2
Example 1: L-shape domain problem
Example 4: Electrostatic micromotor problem
References
Index
414
418
420
42 1
422
424
425
426
427
421
428
430
433
436
439
439
440
44 1
442
444
448
45 1
455
45 8
46 1
468
LIST OF FIGURES
1.1
1.2
1.3
1.4
1.5
1.6
1.7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Jacques Salomon Hadamard ( 1865-1 963).
Isolines of the solution u(z, t) of Burger’s equation.
Johann Peter Gustav Lejeune Dirichlet (1805-1 859).
Maximum principle for the Poisson equation in 2D.
Georg Friedrich Bernhard Riemann (1 826-1866).
Propagation of discontinuity in the solution of the Riemann problem.
Formation of shock in the solution u(z, t) of Burger’s equation.
Boris Grigorievich Galerkin (1 87 1-1945).
Example of a basis function w, of the space V,.
Tridiagonal stiffness matrix S,.
Carl Friedrich Gauss (1777-1855).
Benchmark function f for adaptive numerical quadrature.
Performance of various adaptive Gaussian quadrature rules.
Comparison of adaptive and nonadaptive quadrature.
Piecewise-affine approximate solution to the motivation problem.
6
i
14
27
41
42
44
46
54
55
60
64
64
65
66
xv