Finite Difference Methods on Irregular Networks

The finite difference and finite element methods are powerful tools for the approximate solution of differential equations governing diverse physical phenomena, and there is extensive literature on these discre­ tization methods. In the last two decades, some extensions of the finite difference method to irregular networks have been described and applied to solving boundary value problems in science and engineering. For instance, "box integration methods" have been widely used in electro­ nics. There are several papers on this topic, but a comprehensive study of these methods does not seem to have been attempted. The purpose of this book is to provide a systematic treatment of a generalized finite difference method on irregular networks for solving numerically elliptic boundary value problems. Thus, several disadvan­ tages of the classical finite difference method can be removed, irregular networks of triangles known from the finite element method can be applied, and advantageous properties of the finite difference approxima­ tions will be obtained. The book is written for advanced undergraduates and graduates in the area of numerical analysis as well as for mathematically inclined workers in engineering and science. In preparing the material for this book, the author has greatly benefited from discussions and collaboration with many colleagues who are concerned with finite difference or (and) finite element methods.

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 “discrete” norms.- 4.3.5.A trace theorem and the equivalence of certain grid norms.- 4.4. Green’s formula, inequalities of Friedrichs-Poincaré- 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-Poincaré-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 æ of the principal part of PB-operators.- 5.2.1. The splitting of æ and first estimates.- 5.2.2. The choice of local estimation regions.- 5.2.3. The estimation of de æk and æs.- 5.2.4. The estimation of æu.- 5.2.5. The weakening of the assumptions and conclusions.- 5.3. The error æ 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.