Partial Differential Equations
and the Finite Element Method
PURE AND APPLIED MATHEMATICS
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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

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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

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

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

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

2.6.3 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

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

4.3.10
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

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

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

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

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

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

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

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

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

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

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,

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

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.

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

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.

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

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.

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

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

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

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

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).

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

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.

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

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.

=
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

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

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

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

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