Operator Theory and Numerical Methods, Volume 30

Chapter 1. Elliptic Boundary Value Problems and FEM

1.1 Elliptic Boundary Value Problems

1.2 Ritz-Galerkin Method

1.3 Finite Element Method (FEM)

1.4 Inverse Assumption

1.5 Loo Estimate

1.6 Lp Estimate

1.7 Asymptotic Expansion.


Chapter 2. Semigroup Theory and FEM

2.1 Evolutionary Problems

2.2 Semi-discretization

2.3 Fractional Powers

2.4 Full-discretization

2.5 Inhomogeneous Equation

2.6 Higher Accuracy

2.7 Loo Estimate

2.8 Hyperbolic Equation.


Chapter 3. Evolution Equations and FEM

3.1 Generation Theories

3.2 A Priori Estimates

3.3 Semi-discretization

3.4 Full-discretization

3.5 Alternative Approach.


Chapter 4. Other Methods in Time Discretization

4.1 Rational Approximation of Semigroups

4.2 Multi-step Method

4.3 Product Formula.


Chapter 5. Other Methods in Space Discretization

5.1 Lumping of Mass

5.2 Upwind Finite Elements

5.3 Mixed Finite Elements

5.4 Boundary Element Methods (BEM)

5.5 Charge Simulation Methods (CSM).


Chapter 6. Nonlinear Problems

6.1 Semilinear Elliptic Equations

6.2 Semilinear Parabolic Equations

6.3 Degenerate Parabolic Equations.


Chapter 7. Domain Decomposition Method

7.1 Dirichlet to Neumann (DN) Map

7.2 Dirichlet to Neumann (DN) Iteration

7.3 Dirichlet2 to Neumann2 (DD-NN) Iteration

7.4 Robin to Robin Iteration

7.5 Exterior Problem

7.6 The Stokes System.

Bibliography. Index

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.

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