Numerical methods for elliptic partial differential equations Arnold Reusken

250 Pages·2008·1.51 MB·English

Numerical methods for
elliptic partial differential equations
Arnold Reusken
Preface
This is a book on the numerical approximation of partial differential equations. On the next
page we give an overview of the structure of this book:
2
Elliptic boundary value problems (chapter 1):
Poisson equation: scalar, symmetric, elliptic.
•
Convection-diffusion equation: scalar, nonsymmetric,
•
singularly perturbed.
Stokes equation: system, symmetric, indefinite.
•
Weak formulation (chapter 2)
Basic principles (chapter 3);
•
application to Poisson equation.
Finite element
Streamline-diffusion FEM (chapter 4);
method −→ •
application to convection-diffusion eqn.
FEM for Stokes equation (chapter 5).
•
Basics on linear iterative methods
•
(chapter 6).
Preconditioned CG method (chapter 7);
•
application to Poisson equation.
Krylov subspace methods (chapter 8);
Iterative methods •
−→ application to convection-diffusion eqn.
Multigrid methods (chapter 9).
•
Iterative methods for saddle-point
•
problems (chapter 10);
application to Stokes equation.
A posteriori error estimation (chapter ).
Adaptivity •
−→
Grid refinement techniques (chapter ).
•
3
4
Contents
1 Introduction to elliptic boundary value problems 9
1.1 Preliminaries on function spaces and domains . . . . . . . . . . . . . . . . . . . . 9
1.2 Scalar elliptic boundary value problems . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Existence, uniqueness, regularity . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 The Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Weak formulation 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 The spaces Wm(Ω) based on weak derivatives . . . . . . . . . . . . . . . . 23
2.2.2 The spaces Hm(Ω) based on completion . . . . . . . . . . . . . . . . . . . 25
2.2.3 Properties of Sobolev spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 General results on variational formulations . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Minimization of functionals and saddle-point problems . . . . . . . . . . . . . . . 43
2.5 Variational formulation of scalar elliptic problems . . . . . . . . . . . . . . . . . . 45
2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.2 Elliptic BVP with homogeneous Dirichlet boundary conditions . . . . . . 46
2.5.3 Other boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.4 Regularity results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.5 Riesz-Schauder theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6 Weak formulation of the Stokes problem . . . . . . . . . . . . . . . . . . . . . . . 56
2.6.1 Proof of the inf-sup property . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.6.2 Regularity of the Stokes problem . . . . . . . . . . . . . . . . . . . . . . . 60
2.6.3 Other boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Galerkin discretization and finite element method 63
3.1 Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Examples of finite element spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Simplicial finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.2 Rectangular finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Approximation properties of finite element spaces . . . . . . . . . . . . . . . . . . 68
3.4 Finite element discretization of scalar elliptic problems . . . . . . . . . . . . . . . 75
3.4.1 Error bounds in the norm . . . . . . . . . . . . . . . . . . . . . . . . 75
1
k·k
3.4.2 Error bounds in the norm . . . . . . . . . . . . . . . . . . . . . . . 77
L2
k·k
3.5 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.1 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5
3.6 Isoparametric finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7 Nonconforming finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Finite element discretization of a convection-diffusion problem 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 A variant of the Cea-lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 A one-dimensional hyperbolic problem and its finite element discretization . . . . 93
4.4 The convection-diffusion problem reconsidered. . . . . . . . . . . . . . . . . . . . 100
4.4.1 Well-posedness of the continuous problem . . . . . . . . . . . . . . . . . . 101
4.4.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.3 Stiffness matrix for the convection-diffusion problem . . . . . . . . . . . . 113
5 Finite element discretization of the Stokes problem 115
5.1 Galerkin discretization of saddle-point problems . . . . . . . . . . . . . . . . . . . 115
5.2 Finite element discretization of the Stokes problem . . . . . . . . . . . . . . . . . 117
5.2.1 Error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.2 Other finite element spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Linear iterative methods 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Basic linear iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Convergence analysis in the symmetric positive definite case . . . . . . . . . . . . 134
6.4 Rate of convergence of the SOR method . . . . . . . . . . . . . . . . . . . . . . . 137
6.5 Convergence analysis for regular matrix splittings . . . . . . . . . . . . . . . . . . 140
6.5.1 Perron theory for positive matrices . . . . . . . . . . . . . . . . . . . . . . 141
6.5.2 Regular matrix splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.6 Application to scalar elliptic problems . . . . . . . . . . . . . . . . . . . . . . . . 146
7 Preconditioned Conjugate Gradient method 151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Conjugate Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Introduction to preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4 Preconditioning based on a linear iterative method . . . . . . . . . . . . . . . . . 161
7.5 Preconditioning based on incomplete LU factorizations . . . . . . . . . . . . . . . 162
7.5.1 LU factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.5.2 Incomplete LU factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5.3 Modified incomplete Cholesky method . . . . . . . . . . . . . . . . . . . . 169
7.6 Problem based preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.7 Preconditioned Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . . 170
8 Krylov Subspace Methods 175
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2 The Conjugate Gradient method reconsidered . . . . . . . . . . . . . . . . . . . . 176
8.3 MINRES method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.4 GMRES type of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.5 Bi-CG type of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6
9 Multigrid methods 197
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.2 Multigrid for a one-dimensional model problem . . . . . . . . . . . . . . . . . . . 198
9.3 Multigrid for scalar elliptic problems . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.4 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.4.2 Approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
9.4.3 Smoothing property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
9.4.4 Multigrid contraction number . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.4.5 Convergence analysis for symmetric positive definite problems . . . . . . . 218
9.5 Multigrid for convection-dominated problems . . . . . . . . . . . . . . . . . . . . 223
9.6 Nested Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.8 Algebraic multigrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.9 Nonlinear multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10 Iterative methods for saddle-point problems 229
10.1 Block diagonal preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
10.2 Application to the Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . . . 232
A Functional Analysis 235
A.1 Different types of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
A.2 Theorems from functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 238
B Linear Algebra 241
B.1 Notions from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
B.2 Theorems from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7
8
Chapter 1
Introduction to elliptic boundary
value problems
InthischapterweintroducetheclassicalformulationofscalarellipticproblemsandoftheStokes
equations. Some results known from the literature on existence and uniqueness of a classical
solution will be presented. Furthermore, we briefly discuss the issue of regularity.
1.1 Preliminaries on function spaces and domains
The boundary value problems that we consider in this book will be posed on domains Ω Rn,
⊂
n = 1,2,3. In the remainder we always assume that
Ω is open, bounded and connected.
Moreover, theboundaryofΩshouldsatisfycertainsmoothnesconditionsthatwillbeintroduced
in this section. For this we need so-called H¨older spaces.
By Ck(Ω), k N, we denote the space of functions f :Ω R for which all (partial) derivatives
∈ →
∂ν f
Dνf := | | , ν = (ν ,...,ν ), ν = ν +...+ν ,
∂xν1...∂xνn 1 n | | 1 n
1 n
of order ν k are continuous functions on Ω. Thespace Ck(Ω¯), k N, consists of all functions
| | ≤ ∈
in Ck(Ω) C(Ω¯) for which all derivatives of order k have continuous extensions to Ω¯.
∩ ≤
Since Ω¯ is compact, the functional
f max max Dνf(x) = max Dνf =: f
→ ν k x Ω¯ | | ν kk k∞,Ω¯ k kCk(Ω¯)
| |≤ ∈ | |≤
defines a norm on Ck(Ω¯). The space (Ck(Ω¯),k·kCk(Ω¯)) is a Banach space (cf. Appendix A.1).
Note that f max Dνf does not define a norm on Ck(Ω).
ν k ,Ω
→ | |≤ k k∞
For f :Ω R we define its support by
→
supp(f) := x Ω f(x) = 0 .
{ ∈ | 6 }
The space Ck(Ω), k N, consists of all functions in Ck(Ω) which have a compact support in
0 ∈
Ω, i.e., supp(f) Ω. The functional f max Dνf defines a norm on Ck(Ω), but
⊂ → |ν|≤kk k∞,Ω 0
9
(Ck(Ω), ) is not a Banach space.
0 k·kCk(Ω)
For a compact set D Rn and λ (0,1] we introduce the quantity
⊂ ∈
f(x) f(y)
[f]λ,D := sup{ | x −y λ | | x,y ∈ D, x6= y} for f :D → R.
k − k
We write f C0,λ(Ω¯) and say that f is Ho¨lder continuous in Ω¯ with exponent λ if [f] < .
∈ λ,Ω¯ ∞
A norm on the space C0,λ(Ω¯) is defined by
f f +[f] .
→ k kC(Ω¯) λ,Ω¯
We write f C0,λ(Ω) and say that f is Ho¨lder continuous in Ω with exponent λ if for arbitrary
∈
compact subsets D Ω the property [f] < holds. An important special case is λ = 1: the
λ,D
⊂ ∞
space C0,1(Ω¯) [or C0,1(Ω)] consists of all Lipschitz continuous functions on Ω¯ [Ω].
The space Ck,λ(Ω¯) [Ck,λ(Ω)], k N, λ (0,1], consists of those functions in Ck(Ω¯) [Ck(Ω)]
∈ ∈
for which all derivatives Dνf of order ν = k are elements of C0,λ(Ω¯) [C0,λ(Ω)]. On Ck(Ω¯) we
| |
define a norm by
f f + [Dαf] .
→ k kCk(Ω¯) λ,Ω¯
α=k
|X|
Note that
Ck,λ(Ω¯) Ck(Ω¯) for all k N, λ (0,1],
⊂ ∈ ∈
Ck,λ2(Ω¯) Ck,λ1(Ω¯) for all k N, 0 < λ1 λ2 1 ,
⊂ ∈ ≤ ≤
and similarly with Ω¯ replaced by Ω. We use the notation Ck,0(Ω¯):= Ck(Ω¯) [Ck,0(Ω):= Ck(Ω)].
Remark 1.1.1 The inclusion Ck+1(Ω¯) Ck,λ(Ω¯), λ (0,1], is in general not true. Consider
⊂ ∈
n = 2 and Ω = (x,y) 1< x < 1, 1 < y < x . The function
{ | − − | |}
(signx)y112p if y > 0,
f(x,y) =
(0 otherwise,
belongs to C1(Ω¯), but f / C0,λ(Ω¯) if λ (3,1]. (cid:3)
∈ ∈ 4
Based on these H¨older spaces we can now characterize smoothness of the boundary ∂Ω.
Definition 1.1.2 For k N, λ [0,1] the property ∂Ω Ck,λ (the boundary is of class Ck,λ)
∈ ∈ ∈
holds if at each point x0 ∂Ω there is a ball B = x Rn x x0 < δ, δ > 0 and a
∈ { ∈ | k − k }
bijection ψ : B E Rn such that
→ ⊂
ψ(B ∩Ω) ⊂ Rn+ := {x ∈ Rn | xn > 0}, (1.1a)
ψ(B ∂Ω) ∂Rn, (1.1b)
∩ ⊂ +
ψ Ck,λ(B), ψ 1 Ck,λ(E). (1.1c)
−
∈ ∈
(cid:3)
For the case n = 2 this is illustrated in Figure ??.
Figure 1
10