Elliptic Differential Equations: Theory and Numerical Treatment (Springer Series in Computational Mathematics, 18)

Cover......Page 1
Springer Series in Computational Mathematics 18......Page 2
Elliptic Differential Equations: Theory and Numerical Treatment......Page 4
Copyright - ISBN: 9783540548225......Page 5
Foreword......Page 6
Table of Contents......Page 8
Notation......Page 14
1.1 Examples......Page 16
1.2 Classification of Second-Order Equations into Types......Page 19
1.3 Type Classification for Systems of First Order......Page 21
1.4 Characteristic Properties of the Different Types......Page 22
2.1 Posing the Problem......Page 27
2.2 Singularity Function......Page 29
2.3 The Mean Value Property and Maximum Principle......Page 32
2.4 Continuous Dependence on the Boundary Data......Page 38
3.1 Posing the Problem......Page 42
3.2 Representation of the Solution by the Green Function......Page 43
3.3 The Green Function for the Ball......Page 49
3.4 The Neumann Boundary Value Problem......Page 50
3.5 The Integral Equation Method......Page 51
4.1 Introduction: The One-Dimensional Case......Page 53
4.2 The Five-Point Formula......Page 55
4.3 M-matrices, Matrix Norms, Positive Definite Matrices......Page 59
4.4 Properties of the Matrix L_h......Page 68
4.5 Convergence......Page 74
4.6 Discretisations of Higher Order......Page 77
4.7.1 One-sided Difference for \partial u / \partial n......Page 80
4.7.2 Symmetric Difference for \partial u / \partial n......Page 85
4.7.3 Symmetric Difference for\partial u / \partial n on an Offset Grid......Page 86
4.7.4 Proof of the Stability Theorem 7......Page 87
4.8.1 Shortley-Weller Approximation......Page 93
4.8.2 Interpolation at Points near the Boundary......Page 98
5.1.1 Posing the Problem......Page 100
5.1.2 Maximum Principle......Page 101
5.1.3 Uniqueness of the Solution and Continuous Dependence......Page 102
5.1.4 Difference Methods for the General Differential Equation of Second Order......Page 105
5.2.1 Formulating the Boundary Value Problem......Page 110
5.2.2 Difference Methods for General Boundary Conditions......Page 113
5.3.1 The Biharmonic Differential Equation......Page 118
5.3.2 General Linear Differential Equations of Order 2m......Page 119
5.3.3 Discretisation of the Biharmonic Differential Equation......Page 120
6.1.1 Normed Spaces......Page 125
6.1.2 Operators......Page 126
6.1.3 Banach Spaces......Page 127
6.1.4 Hilbert Spaces......Page 129
6.2.1 L^2(Ω)......Page 130
6.2.2 H^k(Ω) and H^k_0(Ω)......Page 131
6.2.3 Fourier Transformation and H^k(IR^n)......Page 134
6.2.4 H^s(Ω) for Real s \geq 0......Page 137
6.2.5 Trace and Extension Theorems......Page 138
6.3.1 Dual Space of a Normed Space......Page 145
6.3.2 Adjoint Operators......Page 147
6.3.3 Scales of Hilbert Spaces......Page 148
6.4 Compact Operators......Page 150
6.5 Bilinear Forms......Page 152
7.1 Historical Remarks......Page 159
7.2 Equations with Homogeneous Dirichlet Boundary Conditions......Page 160
7.3 Inhomogeneous Dirichlet Boundary Conditions......Page 165
7.4 Natural Boundary Conditions......Page 167
8.1 The Ritz-Galerkin Method......Page 176
8.2 Error Estimates......Page 182
8.3.1 Introduction: Linear Elements for Ω = (a, b)......Page 186
8.3.2 Linear Elements for Ω ubseteq IR^2......Page 189
8.3.3 Bilinear Elements for Ω ubseteq IR^2......Page 193
8.3.4 Quadratic Elements for Ω ubseteq IR^2......Page 195
8.3.6 Handling of Side Conditions......Page 197
8.4.1 H^1-Estimates for Linear Elements......Page 200
8.4.2 L^2 and H^s Estimates for Linear Elements......Page 205
8.5.1 Error Estimates for Other Elements......Page 208
8.5.2.1 Introduction: The One-Dimensional Biharmonic Equation......Page 209
8.5.2.2 The Two-Dimensional Case......Page 210
8.6 Finite Elements for Non-Polygonal Regions......Page 211
8.7.1 Non-Conformal Elements......Page 214
8.7.2 The Trefftz Method......Page 215
8.7.4 Adaptive Triangulation......Page 216
8.7.6 Superconvergence......Page 217
8.8 Properties of the Stiffness Matrix......Page 218
9.1.1 The Regularity Problem......Page 223
9.1.2 Regularity Theorems for Ω = IR^n......Page 225
9.1.3 Regularity Theorems for Ω = IR^n_+......Page 230
9.1.4 Regularity Theorems for General Ω ubseteq IR^n......Page 234
9.1.5 Regularity for Convex Domains and Domains with Corners......Page 238
9.2.1 Discrete H^1-Regularity......Page 241
9.2.2 Consistency......Page 247
9.2.3 Optimal Error Estimates......Page 253
9.2.4 H^2_h-Regularity......Page 255
10.1.1 Formulation......Page 259
10.1.2 Discretisation......Page 261
10.2.1 The Convection-Diffusion Equation......Page 262
10.2.2 Stable Difference Schemes......Page 264
10.2.3 Finite Elements......Page 266
11.1 Formulation of Eigenvalue Problems......Page 268
11.2.1 Discretisation......Page 269
11.2.2 Qualitative Convergence Results......Page 271
11.2.3 Quantitative Convergence Results......Page 275
11.2.4 Complementary Problems......Page 279
11.3 Discretisation by Difference Methods......Page 282
12.1 Systems of Elliptic Differential Equations......Page 290
12.2.1 Weak Formulation of the Stokes Equations......Page 293
12.2.2 Saddlepoint Problems......Page 294
12.2.3 Existence and Uniqueness of the Solution of a Saddlepoint Problem......Page 297
12.2.4 Solvability and Regularity of the Stokes Problem......Page 300
12.2.5 A V_0-elliptic Variational Formulation of the Stokes Problem......Page 304
12.3.1 Finite-Element Discretisation of a Saddlepoint Problem......Page 305
12.3.2 Stability Conditions......Page 306
12.3.3.1 Stability Criterion......Page 308
12.3.3.2 Finite-Element Discretisations with the Bubble Function......Page 309
12.3.3.3 Stable Discretisations with Linear Elements in V_h......Page 311
12.3.3.4 Error Estimates......Page 312
Bibliography......Page 315
Index......Page 322