Numerical Methods for Grid Equations - A. A. Samarskij

. .

Résumé :
5 The Mathematical Theory of Iterative Methods.- 5.1 Several results from functional analysis.- 5.2 Difference schemes as operator equations.- 5.3 Basic concepts from the theory of iterative methods.- 6 Two-Level Iterative Methods.- 6.1 Choosing the iterative parameters.- 6.2 The Chebyshev two-level method.- 6.3 The simple iteration method.- 6.4 The non-self-adjoint case. The simple iteration method.- 6.5 Sample applications of the iterative methods.- 7 Three-Level Iterative Methods.- 7.1 An estimate of the convergence rate.- 7.2 The Chebyshev semi-iterative method.- 7.3 The stationary three-level method.- 7.4 The stability of two-level and three-level methods relative to a priori data.- 8 Iterative Methods of Variational Type.- 8.1 Two-level gradient methods.- 8.2 Examples of two-level gradient methods.- 8.3 Three-level conjugate-direction methods.- 8.4 Examples of the three-level methods.- 8.5 Accelerating the convergence of two-level methods in the self-adjoint case.- 9 Triangular Iterative Methods.- 9.1 The Gauss-Seidel method.- 9.2 The successive over-relaxation method.- 9.3 Triangular methods.- 10 The Alternate-Triangular Method.- 10.1 The general theory of the method.- 10.2 Boundary-value difference problems for elliptic equations in a rectangle.- 10.3 The alternate-triangular method for elliptic equations in arbitrary regions.- 11 The Alternating-Directions Method.- 11.1 The alternating-directions method in the commutative case.- 11.2 Sample applications of the method.- 11.3 The alternating-directions method in the general case.- 12 Methods for Solving Equationswith Indefinite and Singular Operators.- 12.1 Equations with real indefinite operators.- 12.2 Equations with complex operators.- 12.3 General iterative methods for equations with singular operators.- 12.4Special methods.- 13 Iterative Methods for Solving Non-Linear Equations.- 13.1 Iterative methods. The general theory.- 13.2 Methods for solving non-linear difference schemes.- 14 Example Solutions of Elliptic Grid Equations.- 14.1 Methods for constructing implicit iterative schemes.- 14.3 Systems of elliptic equations.- 14.4 Methods for solving elliptic equations in irregular regions.- 15 Methods for Solving Elliptic Equationsin Curvilinear Orthogonal Coordinates.- 15.1 Posing boundary-value problems for differential equations.- 15.2 The solution of difference problems in cylindrical coordinates.- 15.3 Solution of difference problems in polar coordinate systems.- Appendices.- Construction of the minimax polynomial.- Translator's note....