Finite Difference Methods on Irregular Networks: A Generalized Approach to Second Order Elliptic Problems. (1991)

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.- 2.2.1. Networks of triangles and rectangles.- 2.2.2. Locally irregular networks.- 2.3. Secondary networks and boxes.- 2.3.1. General remarks.- 2.3.2. Secondary networks via the perpendicular bisectors (PB).- 2.3.3. Secondary networks via the medians (MD).- 3. Construction of Finite Difference Approximations.- 3.1. The principle of approximation.- 3.2. Finite difference schemes via method (PB).- 3.2,1. The approximation of the balance equations.- 3.2.2. Finite difference schemes of the type (PB).- 3.3. Finite difference schemes via method (MD).- 3.3.1. Quadrature formulas for the balance equations.- 3.3.2. Finite difference quotients.- 3.3.3. Finite difference schemes of the type (MD).- 4. Analytical and Matrix Properties of the Difference Operators Ah.- 4.1. General remarks and notations.- 4.2. Monotonicity and other matrix properties.- 4.2.1. Operators Ah and matrices via method (PB).- 4.2.2. MD-operators Ah in comparison with PB-operators.- 4.3. Scalar products, norms and a trace theorem.- 4.3.1. Notations for scalar products and norms.- 4.3.2. ?-?h-equivalence for norms of special functions.- 4.3.3. ?-?h-relations for norms of special functions.- 4.3.4. Relations between continuous and norms.- 4.3.5.A trace theorem and the equivalence of certain grid norms.- 4.4. Green's formula, inequalities of Friedrichs-Poincare- type and the positive definiteness of Ah.- 4.4.1. Green's formula, the symmetry and the energy of Ah.- 4.4.2. Inequalities of Priedrichs-Poincare-type.- 4.4.3. The positive definiteness of Ah.- 4.5. A priori estimates for Ah using the W12- and C-norm.- 4.5.1. A priori estimates using the W12-norm.- 4.5.2. Weak imbedding of $$ {\text{W}}_{{\text{2}}}^{{\text{1}}}(\bar{\omega })$$ in $$C(\overline \omega)$$.- 4.5.3. A priori estimates using the C-norm.- 5. Error Estimates and Convergence.- 5.1. Error splitting and approaches to the error estimation.- 5.1.1. Introductory remarks and error splitting for PB-schemes.- 5.1.2. Two kinds of error splitting for MD-schemes.- 5.1.3. A priori estimates of the error.- 5.1.4. Convergence for classical solutions of $$C^1 (\overline \Omega)$$-type (1 ? 2).- 5.1.5. Convergence for generalized solutions of $$W_2^1 (\Omega )$$-type (1 ? 2).- 5.2. The error ae of the principal part of PB-operators.- 5.2.1. The splitting of ae and first estimates.- 5.2.2. The choice of local estimation regions.- 5.2.3. The estimation of de aek and aes.- 5.2.4. The estimation of aeu.- 5.2.5. The weakening of the assumptions and conclusions.- 5.3. The error ae of the principal part of MD-operators.- 5.3.1. Splittings and estimates via centres of gravity.- 5.3.2. Splittings and estimates via mesh midpoints.- 5.4. The error ?N for PB- MD-schemes.- 5.4.1. ?N -estimates for grid points x ? ?.- 5.4.2. ?N -estimates for grid points x ? ?23.- 5.4.3. The modification and weakening of some assumptions.- 5.5. Convergence for W22(?)-solutions.- 5.5.1. Convergence in the discrete $$W_2^2 (\Omega )$$>-norm.- 5.5.2. Convergence in the discrete C-norm.- 6. Finite Difference Schemes for Nonsymmetric Problems.- 6.1. Construction of finite difference approximations.- 6.1.1. Boundary value problem and balance equations.- 6.1.2. Upwind difference quotients.- 6.1.3. Further difference quotients and the FDSs $${\text{A}}_{{\text{h}}}^{{\text{b}}}{\text{y = }}{{\text{F}}_{{\text{h}}}}$$.- 6.2. Properties of the difference operators $${\text{A}}_{{\text{h}}}^{{\text{b}}}$$.- 6.2.1. Monotonicity and a priori estimates.- 6.2.2. Positive definiteness and a priori estimates.- 6.3. The error convection term.- 6.3.1. Error splitting and estimates.- 6.3.2. The approach to estimating ?k for $$ u \in W_{2}^{1}\left( \Omega \right)\left( {1 \geqslant 2} \right) $$.- 6.3.3. Local and global bounds of ?k.- 6.4. Convergence for $${C^{1}}\left( {\bar{\Omega }} \right) - $$ and $${\text{W}}_{{\text{2}}}^{{\text{1}}}(\Omega ) - $$-solutions (1 ? 2).- 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.