partial differential equations and the finite element method - pave1 solin

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
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ALLEN
111,
DAVID A.
COX,
PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list
of
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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
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Library
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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
13
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
30
30
31
32
33
33
34
34
35
36
36

38
39
41
43
44
45
45
46
49
50
51
51
51
52
52
53
54
55
56
51
59
59
61
63
65
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
67
69
70
71
74
76
77
79
82
84
84
84
86
88
89
89
89
91
92
93
93
94
96
99
102

103
103
1
04
106
107
108
110
113
114
115
116
119
120
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
126
127
129
131
133
134
135
137
137
139

139
139
142
142
143
148
149
152
154
157
160
162
166
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
167
168
168
168
169
170
171
172
173
175
177
179

182
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
186
187
188
191
193
194
197
197
200
202
205
209
210
210
212
214
214
216

216
218
220
220
222
225
226
228
228
23
1
233
236
236
238
240
242
242
243
246
248
250
254
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
255
255
256
257
259
260

265
266
269
270
270
212
273
274
275
275
276
277
279
279
28
1
28
1
283
283
284
285
285
286
287
288
289
289
290
290

293
300
30
1
302
309
314
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
317
318
31
9
320
320
32
1
323

326
327
328
332
337
339
34
1
343
345
348
348
352
355
357
36
1
363
366
37
1
375
380
3 84
386
389
389
394
395
399

40
1
405
407
408
409
412
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
xvi
LIST
OF
FIGURES
2.9
2.10
2.1
1
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29
2.30

2.3
1
2.32
2.33
3.1
3.2
3.3
3.4
3.5
Quadratic approximate solution
to
the motivation problem.
Joseph-Louis Lagrange
(1736-1 8 13).
Pafnuty Lvovich Chebyshev
(1
821-1894).
Comparison of the Gauss-Lobatto and Chebyshev points.
Lagrange-Gauss-Lobatto nodal shape functions,
p
=
2.
Lagrange-Gauss-Lobatto nodal shape functions,
p
=
3.
Lagrange-Gauss-Lobatto nodal shape functions,
p
=
4.

Lagrange-Gauss-Lobatto nodal shape functions,
p
=
5.
Lowest-order Lobatto hierarchic shape functions.
HA
-orthonormal (Lobatto) hierarchic shape functions,
p
=
2,3.
H&orthonormal (Lobatto) hierarchic shape functions,
p
=
4.5.
Hd-orthonormal (Lobatto) hierarchic shape functions,
p
=
6,7.
H&orthonormal (Lobatto) hierarchic shape functions,
p
=
8.9.
Piecewise-quadratic vertex basis function.
Condition number vs. performance of an iterative matrix solver.
Condition number of the stiffness matrix for various
p.
Condition number of the mass matrix for various
p.
Stiffness matrix for the Lobatto hierarchic shape functions.
Example of a Dirichlet lift function.

Dirichlet lift for combined boundary conditions
(2.79).
Best approximation
gh,p
E
v,,p
of the function
g
E
v.
Projection-based interpolation.
Graphical interpretation of the projection problem
(2.94).
67
70
71
72
73
73
73
73
75
75
75
75
75
76
85
87
87

88
90
92
95
96
98
Error factor
&(x)
for equidistributed nodal points,
p
=
4,7,10,
and
13.
100
Error factor
&(x)
for Chebyshev nodal points,
p
=
4,7,10
and 13.
101
Example of a nonunisolvent nodal finite element.
107
Q1-element on the reference domain
Kq.
108
Q1-element on a physical mesh quadrilateral.
109

Richard Courant
(1888-1972).
111
PI-element
on
the reference domain.
I11
3.6
3.7
3.8
3.9
3.10
4.1
4.2
4.3
4.4
4.5
4.6
4.1
4.8
4.9
4.10
4.1
1
4.12
4.13
4.14
4.15
4.16
4.17

4.18
4.19
4.20
4.2
1
4.22
4.23
4.24
4.25
LIST
OF
FIGURES
xvii
PI-element on a physical mesh triangle.
112
1
I6
118
119
121
The domain
R,
its boundary
dll,
and the unit outer normal vector
v
to
dR.
126
Example of a nodal interpolant on the Q1-element.

Example of a global interpolant that is continuous.
Example of a discontinuous global interpolant.
Example of a pair of nonequivalent elements.
Polygonal approximation
fitL
of the domain
(2.
Generally
Rh
#
R
Example of triangular, quadrilateral, and hybrid meshes.
Vertex basis functions on P1/Q1-meshes.
Orientation of edges on the reference quadrilateral
Kq.
Nodal shape functions on the Q2-element; vertex functions.
Nodal shape functions on the Q2-element; edge functions.
Nodal shape functions on the Q2-element; bubble function.
Nodal shape functions on the Q'-element; vertex functions.
Nodal shape functions on the Q3-element; edge functions
p
=
2.
Nodal shape functions on the Q3-element; edge functions
p
=
3.
Nodal shape functions on the Q3-element; bubble functions.
Gauss-Lobatto points
in

a physical mesh quadrilateral.
The Fekete points in
zt,
p
=
1,2,.
.
.
,15.
Orientation of edges on the reference triangle
Kt.
Nodal basis of the P2-element; vertex functions.
Nodal basis of the P2-element; edge functions.
Nodal basis of the P3-element; vertex functions.
Nodal basis of the P3-element; edge functions
(p
=
2).
Nodal basis of the P3-element; edge functions
(p
=
3).
Nodal basis of the P'-element; bubble function.
Mismatched nodal points on Q'/Q2-element interface.
Example of a vertex element patch.
Example of an edge element patch.
Examples of bubble functions.
127
128
130

142
145
145
145
146
146
146
146
147
151
152
153
153
154
154
154
154
155
155
156
157
xviii
LIST
OF
FIGURES
4.26
5.1
5.2
5.3
6.1

6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.1
1
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
Enumeration of basis functions.
Example of a stiff ODE problem.
Carle David Tolme Runge (1 856-1927).
Stability domain of the explicit Euler method.

Bending of a prismatic beam; initial and deformed configurations.
Strain induced by the deflection of a beam.
Clamped beam boundary conditions.
Simply supported beam boundary conditions.
Cantilever beam boundary conditions.
Cubic shape functions representing function values.
Cubic shape functions representing the derivatives.
Fourth-order vertex functions representing function values.
Fourth-order bubble function representing function values.
Fourth-order vertex functions representing derivatives.
Hi-orthonormal hierarchic shape functions
0,
=
4,5).
H$orthonormal hierarchic shape functions
0,
=
6,7).
Hi-orthonormal hierarchic shape functions
(JJ
=
8,9).
H&orthonormal hierarchic shape functions
(JJ
=
10,ll).
Conditioning comparison in the Hi-product.
Conditioning comparison in the HA-product
Two equivalent types of cubic Hermite elements.
Nodal basis of the cubic Hermite element; vertex functions.

Nodal basis of the cubic Hermite element; bubble function.
Nodal basis of the cubic Hermite element; vertex functions
Nodal basis of the cubic Hermite element; vertex functions
(i3/&2).
Fourth- and fifth-order Hermite-Fekete elements on
Kt
.
163
178
179
190
210
21
1
213
213
213
219
219
22 1
22
1
222
224
224
224
224
225
226
236

238
238
238
238
239
The transversal force, shear resultant, and bending and twisting moments.246
Hypothesis
(P5)
in the Kirchhoff plate model. 247
Clamped, simply supported, and traction boundary conditions. 250
BabuSka’s paradox of thin plates.
254
6.27
6.28
6.29
6.30
6.3
1
6.32
6.33
6.34
6.35
6.36
7.1
7.2
7.3
7.4
7.5
7.6
7.7

7.8
7.9
A.
1
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.10
A.11
LIST
OF
FIGURES
Twenty-one
DOF
on the lowest-order (quintic) Argyris triangle.
Conformity of Argyris elements.
Nodal basis of the quintic Argyris element; part 1.
Nodal basis of the quintic Argyris element; part 2.
Nodal basis
of
the quintic Argyris element; part
3.
Nodal basis of the quintic Argyris element; part 4.
Nodal basis of the quintic Argyris element; part 5.
Nodal basis

of
the quintic Argyris element; part
6.
Nodal basis
of
the quintic Argyris element; part 7.
The sixth- and seventh-order Argyris-Fekete elements on
Kt
.
Parameterization of a smooth curve and its derivative.
James Clerk Maxwell (1831-1879).
Electric field on a media interface.
Magnetic field on a media interface.
Current field on a media interface.
Internal interface separating regions with different material properties.
Orientation of the edges on the reference domain
Kt
.
Affine transformation
XK
:
Kt
+
K.
Element patch
Se(j)
corresponding to an interior mesh edge
s3.
Structure of linear spaces discussed in this chapter.
Example of a set which is not a linear space.

Subspace
W
corresponding to the vector
w
=
(2,
l)T.
Example
of
intersection of subspaces.
Example of union of subspaces.
Unique decomposition of a vector in a direct sum of subspaces.
Linear operator in
R2
(rotation of vectors).
Canonical basis of
R3.
Basis
B
=
{q,
~2,213).
Examples of unit open balls
B(0,l)
in
V
=
R2.
Open ball in a polynomial space equipped with the maximum norm.
xix

255
257
258
258
258
258
258
259
259
266
27 1
272
277
278
278
286
302
306
309
319
322
323
324
325
327
336
338
344
355
355

XX
LIST
OF
FIGURES
A.12
A.13
A.14
A.15
A.16
A.17
A.18
A. 19
A.20
A.21
A.22
A.23
A.24
A.25
A.26
A.27
A.28
A.29
A.30
A.3
1
A.32
A.33
A.34
A.35
A.36

B.
1
B.2
B.3
B.4
8.5
Open ball in a polynomial space equipped with the integral norm.
Space where the derivative operator
is
not continuous.
Set closed in the maximum norm but open in the integral norm.
Nonconvergent Cauchy sequence in the space
C(
(0,Zl).
Stefan Banach (1892-1945).
Approximate calculation
of
a square root.
Solution of the equation
x'i
+
z
-
1
=
0
via fixed point iteration.
Solution of the equation
n:
-

cos(z)
=
0
via local fixed point iteration
Henri Leon Lebesgue
(
1
875-1 94
1
).
Function which
is
not integrable by means of the Riemann integral.
Otto Ludwig Holder
(1
859-1
937).
Hermann Minkowski
(1
864-1 909).
Structure
of
LP-spaces on an open bounded set.
Example of a sequence converging out of
C(
-
1,l).
David Hilbert
(1
862-1943).

First five Legendre polynomials
Lo, L1,.
. .
,
L4.
Jean Baptiste Joseph Fourier (1768-1 830).
Fourier series of the discontinuous function
g
E
L2(0,
27r).
Frigyes Riesz
(
1
880-
1956).
Parallelogram
ABCD
in
R2.
Sergei Lvovich Sobolev
(
1908- 1989).
An open bounded set which
(a)
is
and (b)
is
not a domain.
Bounded set with infinitely long boundary.

Illustration of the Lipschitz-continuity of
dn.
The functions
cp
and
$.
Structure of the modular
EM
system HERMES.
Geometry of the L-shape domain.
Approximate solution
7Lh.p
of
the L-shape domain problem.
Detailed view of
JVU~~,,~
at
the reentrant corner.
The hp-mesh, global view.
356
359
366
367
368
370
373
374
375
377
38

1
382
384
385
394
398
399
40
1
405
409
412
413
413
414
415
440
444
445
445
446
LIST
OF
FIGURES
xxi
B.6
B.7
B.8
B.9
B.10

B.ll
B.12
B.13
B.14
B.15
B.16
B.17
B.18
B.19
B.20
B.21
B.22
B.23
B.24
B.25
The hp-mesh, details of the reentrant comer.
A-posteriori error estimate for
?Lh,p.
details of the reentrant comer.
Geometry of the insulator problem.
Approximate solution
ptL,p of
the insulator problem.
Details
of
the singularity of
IEh,pl
at the reentrant corner, and the
discontinuity along the material interface.
The hp-mesh, global view.

The hp-mesh, details
of
the reentrant corner.
A-posteriori error estimate for
ph,p,
details
of
the reentrant comer.
Computational domain
of
the cone-sphere problem.
Approximate solution
p)t,p
of the cone-sphere problem.
Details of the singularity
of
IEh,pl
at the tip of the cone.
The hp-mesh, global view.
The hp-mesh, details of the tip
of
the cone
A-posteriori error estimate for
y~~,~,
details of the reentrant corner.
Geometry of the micromotor problem.
Approximate solution
ph,+
of the micromotor problem.
The hp-mesh.

Approximate solution to the diffraction problem.
The hp-mesh consisting
of
hierarchic edge elements.
The mesh consisting of the lowest-order (Whitney) edge elements.
446
447
448
449
449
449
450
450
45
1
452
452
453
453
454
455
456
457
459
459
460
LIST
OF
TABLES
2.1

2.2
2.3
2.4
2.5
4.1
4.2
4.3
4.4
4.5
4.6
4.1
4.8
5.1
5.2
Gaussian quadrature on
KO,
order
2k
-
1
=
3.
Gaussian quadrature on
K,,
order
2k
-
1
=
5.

Gaussian quadrature on
K,,
order
2k
-
1
=
7.
Gaussian quadrature on
K,,
order
2k
-
1
=
9.
Gaussian quadrature on
Ka,
order
2k
-
1
=
11.
Gaussian quadrature on
Kt,
order
p
=
1.

Gaussian quadrature on
Kt,
order
p
=
2.
Gaussian quadrature on
Kt
,
order
p
=
3.
Gaussian quadrature on
Kt
,
order
p
=
4.
Gaussian quadrature on
Kt
,
order
p
=
5.
Fekete points in
K,,
p

=
1.
Fekete points in
Kt,
p
=
2.
Approximate Fekete points in
Kt,
p
=
3.
Minimum number of stages
for
a pth-order
RK
method.
Coefficients
of
the Dormand-Prince
RK5(4)
method.
61
62
62
62
62
141
141
141

141
141
150
150
150
182
183
xxiii
xxiv
CONTENTS
B.l
Efficiency comparison of the piecewise-affine FEM and hp-FEM.
447
B.2
Efficiency comparison of the piecewise-affine FEM and hp-EM.
450
B.3
Efficiency comparison of the piecewise-affine FEM and hp-FEM.
454
B.4
Efficiency comparison of the piecewise-affine FEM and hp-FEM.
458
B.5
460
Efficiency comparison
of
the lowest-order and
hp
edge elements.
PREFACE

Rien ne serf de couril;
i1,faut partir
a
point.
Jean de la Fontaine
Many physical processes in nature, whose correct understanding, prediction, and control
are important to people, are described by equations that involve physical quantities together
with their spatial and temporal rates of change
(partial derivatives).
Among such processes
are the weather, flow of liquids, deformation of solid bodies, heat transfer, chemical reac-
tions, electromagnetics, and many others. Equations involving partial derivatives are called
partial diferential equations
(PDEs).
The solutions to these equations are functions, as
opposed to standard algebraic equations whose solutions are numbers.
For
most PDEs we
are not able to find their exact solutions, and sometimes we do not even know whether a
unique solution exists. For these reasons, in most cases the only way to solve PDEs arising
in concrete engineering and scientific problems is to approximate their solutions numeri-
cally. Numerical methods for PDEs constitute an indivisible part of modern engineering
and science.
The most general and efficient tool for the numerical solution of PDEs is the
Finite
element method
(FEM),
which is based on the spatial subdivision of the physical domain
intofinite
elements

(often triangles
or
quadrilaterals in 2D and tetrahedra, bricks,
or
prisms
in 3D), where the solution is approximated via a finite set of polynomial
skape,funcrions.
In this way the original problem
is
transformed into a
discrete problem
for a finite number
of unknown coefficients. It is worth mentioning that rather simple shape functions, such
as affine
or
quadratic polynomials, have been used most frequently in the past due to
their relatively low implementation cost. Nowadays, higher-order elements are becoming
increasingly popular due
to
their excellent approximation properties and capability to reduce
the size of finite element computations significantly.
The higher-order finite element methods, however, require a better knowledge
of
the
underlying mathematics.
In
particular, the understanding of linear algebra and elementary
xxv
xxvi
PREFACE

functional analysis is necessary. In this book we follow the modern trend of building
engineering finite element methods upon a solid mathematical foundation, which can be
traced in several other recent finite element textbooks, as, e.g.,
[
181
(membrane, beam and
plate models),
[29]
(finite element analysis of shells),
or
[83] (edge elements for Maxwell’s
equations).
The contents at
a
glance
This book is aimed at graduate and Ph.1~. students of all disciplines of computational engi-
neering and science. It provides an introduction into the modern theory of partial differential
equations, finite element methods, and their applications. The logical beginning of the text
lies
in Appendix A, which is a course in linear algebra and elementary functional analy-
sis. This chapter is readable with minimum prerequisites and it contains many illustrative
examples. Readers who trust their skills in function spaces and linear operators may skip
Appendix A, but it will facilitate the study of PDEs and finite element methods to all others
significantly.
The core Chapters
14
provide an introduction to the theory of PDEs and finite element
methods. Chapter
5
is devoted to the numerical solution of ordinary differential equations

(ODES) which arise in the semidiscretization of time-dependent PDEs by the most fre-
quently used
Method
of
lines (MOL).
Emphasis is given to higher-order implicit one-step
methods. Chapter
6
deals with Hermite and Argyris elements with application to fourth-
order problems rooted in the bending of elastic beams and plates. Since the fourth-order
problems are less standard than second-order equations, their physical background and
derivation are discussed in more detail. Chapter
7
is a newcomer’s introduction into com-
putational electromagnetics. Explained are basic laws governing electromagnetics in both
their integral and differential forms, material properties, constitutive relations, and interface
conditions. Discussed are potentials and problems formulated in terms of potentials, and
the time-domain and time-harmonic Maxwell’s equations. The concept of NCdClec’s
edge
elements
for the Maxwell’s equations is explained.
Appendix
B
deals with selected algorithmic and programming issues. We present a uni-
versal sparse matrix interface sMatrix which makes it possible to connect multiple sparse
matrix solver packages simultaneously to a finite element solver. We mention the advantages
of separating the finite element technology from the physics represented by concrete PDEs.
Such approach is used in the implementation of a high-performance modular finite element
system HERMES. This software
is

briefly described and applied to several challenging
engineering problems formulated in terms of second-order elliptic PDEs and time-harmonic
Maxwell’s equations. Advantages of higher-order elements are demonstrated.
After studying this introductory text, the reader should be ready to read articles and
monographs on advanced topics including a-posteriori error estimation and automatic adap-
tivity, mixed finite element formulations and saddle point problems, spectral finite element
methods, finite element multigrid methods, hierarchic higher-order finite element methods
(hp-FEM), and others (see, e.g.,
[9,23,69,
1051
and
[
1 1
11). Additional test and homework
problems, along with an errata, will be maintained on my home page.
PAVEL
SOL~N
ACKNOWLEDGMENTS
I
acknowledge with gratitude the assistance and help of many friends, colleagues and
students in the preparation of the manuscript.’ Tom% Vejchodskf (Academy of Sciences of
the Czech Republic) read a significant part of the text and provided me with many corrections
and hints that improved its overall quality. Martin Zitka (Charles University, Prague, and
UTEP) checked Chapter
2
and made numerous useful observations to various other parts of
the text. Invaluable was the expert review
of
the ODE Chapter
5

by Laurent
Jay
(University
of Iowa). The functional-analytic course in Appendix A was reviewed by Volker John
(Universitat des Saarlandes, Saarbrucken) from the point
of
view of a numerical analyst,
and by Osvaldo Mendez (UTEP), who is an expert in functional analysis.
For
numerous
corrections to this part of the text
I
also wish to thank
to
UTEP’s graduate students Svatava
Vyvialova and Francisco Avila.
I
am deeply indebted to Prof. Ivo Doleiel (Czech Technical University and Academy of
Sciences of the Czech Republic), who is a theoretical electrical engineer with lively inter-
est in computational mathematics,
for
providing me over the years with exciting practical
problems
to
solve. Mainly thanks to him
I
learned to appreciate the engineer’s point
of
view. The manuscript emerged from handouts, course notes, homeworks, and tests written
for students. The students along with their interest and excitement were my main sources

of motivation to write this book.
There is
no
way to express all my gratitude to my wife Dagmar for her support, under-
standing, and admirable patience during the two years of my work on the manuscript.
P.
5.
‘The author acknowledges the support
of
the Czech Science Foundation under the Grant
No.
102/05/0629.
xxvii