Operator theory and numerical methods

H. Fujita, N. Saito, T. Suzuki0444504745, 9780444504746, 9780080538020

In accordance with the developments in computation, theoretical studies on numerical schemes are now fruitful and highly needed. In 1991 an article on the finite element method applied to evolutionary problems was published. Following the method, basically this book studies various schemes from operator theoretical points of view. Many parts are devoted to the finite element method, but other schemes and problems (charge simulation method, domain decomposition method, nonlinear problems, and so forth) are also discussed, motivated by the observation that practically useful schemes have fine mathematical structures and the converses are also true. This book has the following chapters: 1. Boundary Value Problems and FEM. 2. Semigroup Theory and FEM. 3. Evolution Equations and FEM. 4. Other Methods in Time Discretization. 5. Other Methods in Space Discretization. 6. Nonlinear Problems. 7. Domain Decomposition Method.

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Table of contents :
Cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Contents……Page 7
1.1 Elliptic Boundary Value Problems……Page 9
1.2 Ritz-Galerkin Method……Page 15
1.3 Finite Element Method (FEM)……Page 18
1.4 Inverse Assumption……Page 22
1.5 $L^infty$ Estimate……Page 27
1.6 $L^p$ Estimate……Page 35
1.7 Asymptotic Expansion……Page 43
2.1 Evolutionary Problems……Page 55
2.2 Semi-discretization……Page 61
2.3 Fractional Powers……Page 66
2.4 Full-discretization……Page 69
2.5 Inhomogeneous Equation……Page 76
2.6 Higher Accuracy……Page 78
2.7 $L^infty$ Estimate……Page 85
2.8 Hyperbolic Equation……Page 91
3.1 Generation Theories……Page 103
3.2 A Priori Estimates……Page 116
3.3 Semi-discretization……Page 126
3.4 Full-discretization……Page 134
3.5 Alternative Approach……Page 139
4.1 Rational Approximation of Semigroups……Page 153
4.2 Multi-step Method……Page 161
4.3 Product Formula……Page 173
5.1 Lumping of Mass……Page 179
5.2 Upwind Finite Elements……Page 189
5.3 Mixed Finite Elements……Page 190
5.4 Boundary Element Method (BEM)……Page 198
5.5 Charge Simulation Method (CSM)……Page 203
6.1 Semilinear Elliptic Equations……Page 215
6.2 Semilinear Parabolic Equations……Page 219
6.3 Degenerate Parabolic Equations……Page 225
7.1 Dirichlet to Neumann (DN) Map……Page 251
7.2 Dirichlet to Neumann (DN) Iteration……Page 254
7.3 Dirichlet$^2$ to Neumann$^2$ (DD-NN) Iteration……Page 263
7.4 Robin to Robin (RR) Iteration……Page 265
7.5 Exterior Problem……Page 266
7.6 The Stokes System……Page 269
Bibliography……Page 283
Index……Page 311