Author | Heinrich, Bernd. author |
---|---|
Title | Finite Difference Methods on Irregular Networks [electronic resource] : A Generalized Approach to Second Order Elliptic Problems / by Bernd Heinrich |
Imprint | Basel : Birkhรคuser Basel, 1987 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-7196-9 |
Descript | 206 p. online resource |
1. Introduction -- 1.1. Preliminary remarks -- 1.2. Scope of monograph -- 1.3. Plan of monograph, comments -- 2. Boundary Value Problems and Irregular Networks -- 2.1. A class of elliptic problems -- 2.2. Irregular networks -- 2.3. Secondary networks and boxes -- 3. Construction of Finite Difference Approximations -- 3.1. The principle of approximation -- 3.2. Finite difference schemes via method (PB) -- 3.3. Finite difference schemes via method (MD) -- 4. Analytical and Matrix Properties of the Difference Operators Ah -- 4.1. General remarks and notations -- 4.2. Monotonicity and other matrix properties -- 4.3. Scalar products, norms and a trace theorem -- 4.4. Greenโs formula, inequalities of Friedrichs-Poincarรฉ- type and the positive definiteness of Ah -- 4.5. A priori estimates for Ah using the W12- and C-norm -- 5. Error Estimates and Convergence -- 5.1. Error splitting and approaches to the error estimation -- 5.2. The error รฆ of the principal part of PB-operators -- 5.3. The error รฆ of the principal part of MD-operators -- 5.4. The error ?N for PB- MD-schemes -- 5.5. Convergence for W22(?)-solutions -- 6. Finite Difference Schemes for Nonsymmetric Problems -- 6.1. Construction of finite difference approximations -- 6.2. Properties of the difference operators $${\text{A}}_{{\text{h}}}̂{{\text{b}}}$$ -- 6.3. The error convection term -- 7. Concluding Remarks -- Appendices -- 1. Appendix DI: Relations of Differential and Integral calculus, norms -- 2. Appendix ES: Estimation of functionals on Sobolev spaces -- 3. Appendix EX: Extension of functions -- 4. Appendix GE: Some relations of geometry -- 5. Appendix IM: Imbedding and trace theorems -- 6. Appendix TR: Affine transformations of coordinates and functional -- References -- List of Figures -- Abbreviations -- Notations