Numerical Methods for Grid Equations

Inhaltsverzeichnis

• 5 The Mathematical Theory of Iterative Methods.
• 5.1 Several results from functional analysis.
• 5.1.1 Linear spaces.
• 5.1.2 Operators in linear normed spaces.
• 5.1.3 Operators in a Hilbert space.
• 5.1.4 Functions of a bounded operator.
• 5.1.5 Operators in a finite-dimensional space.
• 5.1.6 The solubility of operator equations.
• 5.2 Difference schemes as operator equations.
• 5.2.1 Examples of grid-function spaces.
• 5.2.2 Several difference identities.
• 5.2.3 Bounds for the simplest difference operators.
• 5.2.4 Lower bounds for certain difference operators.
• 5.2.5 Upper bounds for difference operators.
• 5.2.6 Difference schemes as operator equations in abstract spaces.
• 5.2.7 Difference schemes for elliptic equations with constant coefficients.
• 5.2.8 Equations with variable coefficients and with mixed derivatives.
• 5.3 Basic concepts from the theory of iterative methods.
• 5.3.1 The steady state method.
• 5.3.2 Iterative schemes.
• 5.3.3 Convergence and iteration counts.
• 5.3.4 Classification of iterative methods.
• 6 Two-Level Iterative Methods.
• 6.1 Choosing the iterative parameters.
• 6.1.1 The initial family of iterative schemes.
• 6.1.2 The problem for the error.
• 6.1.3 The self-adjoint case.
• 6.2 The Chebyshev two-level method.
• 6.2.1 Construction of the set of iterative parameters.
• 6.2.2 On the optimality of the a priori estimate.
• 6.2.3 Sample choices for the operator D.
• 6.2.4 On the computational stability of the method.
• 6.2.5 Construction of the optimal sequence of iterative parameters.
• 6.3 The simple iteration method.
• 6.3.1 The choice of the iterative parameter.
• 6.3.2 An estimate for the norm of the transformation operator.
• 6.4 The non-self-adjoint case. The simple iteration method.
• 6.4.1 Statement of the problem.
• 6.4.2 Minimizing the norm of the transformation operator.
• 6.4.3 Minimizing the norm of the resolving operator.
• 6.4.4 The symmetrization method.
• 6.5 Sample applications of the iterative methods.
• 6.5.1 A Dirichlet difference problem for Poisson’s equation in a rectangle.
• 6.5.2 A Dirichlet difference problem for Poisson’s equation in an arbitrary region.
• 6.5.3 A Dirichlet difference problem for an elliptic equation with variable coefficients.
• 6.5.4 A Dirichlet difference problem for an elliptic equation with mixed derivatives.
• 7 Three-Level Iterative Methods.
• 7.1 An estimate of the convergence rate.
• 7.1.1 The basic family of iterative schemes.
• 7.1.2 An estimate for the norm of the error.
• 7.2 The Chebyshev semi-iterative method.
• 7.2.1 Formulas for the iterative parameters.
• 7.2.2 Sample choices for the operator D.
• 7.2.3 The algorithm of the method.
• 7.3 The stationary three-level method.
• 7.3.1 The choice of the iterative parameters.
• 7.3.2 An estimate for the rate of convergence.
• 7.4 The stability of two-level and three-level methods relative to a priori data.
• 7.4.1 Statement of the problem.
• 7.4.2 Estimates for the convergence rates of the methods.
• 8 Iterative Methods of Variational Type.
• 8.1 Two-level gradient methods.
• 8.1.1 The choice of the iterative parameters.
• 8.1.2 A formula for the iterative parameters.
• 8.1.3 An estimate of the convergence rate.
• 8.1.4 Optimality of the estimate in the self-adjoint case.
• 8.1.5 An asymptotic property of the gradient methods in the self-adjoint case.
• 8.2 Examples of two-level gradient methods.
• 8.2.1 The steepest-descent method.
• 8.2.2 The minimal residual method.
• 8.2.3 The minimal correction method.
• 8.2.4 The minimal error method.
• 8.2.5 A sample application of two-level methods.
• 8.3 Three-level conjugate-direction methods.
• 8.3.1 The choice of the iterative parameters. An estimate of the convergence rate.
• 8.3.2 Formulas for the iterative parameters. The three-level iterative scheme.
• 8.3.3 Variants of the computational formulas.
• 8.4 Examples of the three-level methods.
• 8.4.1 Special cases of the conjugate-direction methods.
• 8.4.2 Locally optimal three-level methods.
• 8.5 Accelerating the convergence of two-level methods in the self-adjoint case.
• 8.5.1 An algorithm for the acceleration process.
• 8.5.2 An estimate of the effectiveness.
• 8.5.3 An example.
• 9 Triangular Iterative Methods.
• 9.1 The Gauss-Seidel method.
• 9.1.1 The iterative scheme for the method.
• 9.1.2 Sample applications of the method.
• 9.1.3 Sufficient conditions for convergence.
• 9.2 The successive over-relaxation method.
• 9.2.1 The iterative scheme. Sufficient conditions for covergence.
• 9.2.2 The choice of the iterative parameter.
• 9.2.3 An estimate of the spectral radius.
• 9.2.4 A Dirichlet difference problem for Poisson’s equation in a rectangle.
• 9.2.5 A Dirichlet difference problem for an elliptic equation with variable coefficients.
• 9.3 Triangular methods.
• 9.3.1 The iterative scheme.
• 9.3.2 An estimate of the convergence rate.
• 9.3.3 The choice of the iterative parameter.
• 9.3.4 An estimate for the convergence rates of the Gauss-Seidel and relaxation methods.
• 10 The Alternate-Triangular Method.
• 10.1 The general theory of the method.
• 10.1.1 The iterative scheme.
• 10.1.2 Choice of the iterative parameters.
• 10.1.3 A method for finding ? and ?.
• 10.1.4 A Dirichlet difference problem for Poisson’s equation in a rectangle.
• 10.2 Boundary-value difference problems for elliptic equations in a rectangle.
• 10.2.1 A Dirichlet problem for an equation with variable coefficients.
• 10.2.2 A modified alternate-triangular method.
• 10.2.3 A comparison of the variants of the method.
• 10.2.4 A boundary-value problem of the third kind.
• 10.2.5 A Dirichlet difference problem for an equation with mixed derivatives.
• 10.3 The alternate-triangular method for elliptic equations in arbitrary regions.
• 10.3.1 The statement of the difference problem.
• 10.3.2 The construction of an alternate-triangular method.
• 10.3.3 A Dirichlet problem for Poisson’s equation in an arbitrary region.
• 11 The Alternating-Directions Method.
• 11.1 The alternating-directions method in the commutative case.
• 11.1.1 The iterative scheme for the method.
• 11.1.2 The choice of the parameters.
• 11.1.3 A fractionally-linear transformation.
• 11.1.4 The optimal set of parameters.
• 11.2 Sample applications of the method.
• 11.2.1 A Dirichlet difference problem for Poisson’s equation in a rectangle.
• 11.2.2 A boundary-value problem of the third kind for an elliptic equation with separable variables.
• 11.2.3 A high-accuracy Dirichlet difference problem.
• 11.3 The alternating-directions method in the general case.
• 11.3.1 The case of non-commuting operators.
• 11.3.2 A Dirichlet difference problem for an elliptic equation with variable coefficients.
• 12 Methods for Solving Equationswith Indefinite and Singular Operators.
• 12.1 Equations with real indefinite operators.
• 12.1.1 The iterative scheme. The choice of the iterative parameters.
• 12.1.2 Transforming the operator in the self-adjoint case.
• 12.1.3 The iterative method with the Chebyshev parameters.
• 12.1.4 Iterative methods of variational type.
• 12.1.5 Examples.
• 12.2 Equations with complex operators.
• 12.2.1 The simple iteration method.
• 12.2.2 The alternating-directions method.
• 12.3 General iterative methods for equations with singular operators.
• 12.3.1 Iterative schemes in the case of a non-singular operator B.
• 12.3.2 The minimum-residual iterative method.
• 12.3.3 A method with the Chebyshev parameters.
• 12.4 Special methods.
• 12.4.1 A Neumann difference problem for Poisson’s equation in a rectangle.
• 12.4.2 A direct method for the Neumann problem.
• 12.4.3 Iterative schemes with a singular operator B.
• 13 Iterative Methods for Solving Non-Linear Equations.
• 13.1 Iterative methods. The general theory.
• 13.1.1 The simple iteration method for equations with a monotone operator.
• 13.1.2 Iterative methods for the case of a differentiate operator.
• 13.1.3 The Newton-Kantorovich method.
• 13.1.4 Two-stage iterative methods.
• 13.1.5 Other iterative methods.
• 13.2 Methods for solving non-linear difference schemes.
• 13.2.1 A difference scheme for a one-dimensional elliptic quasi-linear equation.
• 13.2.2 The simple iteration method.
• 13.2.3 Iterative methods for quasi-linear elliptic difference equations in a rectangle.
• 13.2.4 Iterative methods for weakly-nonlinear equations.
• 14 Example Solutions of Elliptic Grid Equations.
• 14.1 Methods for constructing implicit iterative schemes.
• 14.1.1 The regularizer principle in the general theory of iterative methods.
• 14.1.2 Iterative schemes with a factored operator.
• 14.1.3 A method for implicity inverting the operator B (a two-stage method).
• 14.2 Examples of solving elliptic boundary-value problems.
• 14.2.1 Direct and iterative methods.
• 14.2.2 A high-accuracy Dirichlet difference problem in the multi-dimensional case.
• 14.2.3 A boundary-value problem of the third kind for an equation with mixed derivatives in a rectangle.
• 14.2.4 Iterative methods for solving a difference problem.
• 14.3 Systems of elliptic equations.
• 14.3.1 A Dirichlet problem for systems of elliptic equations in a p-dimensional parallelipiped.
• 14.3.2 A system of equations from elasticity theory.
• 14.4 Methods for solving elliptic equations in irregular regions.
• 14.4.1 Difference problems in regions of complex form and methods for their solution.
• 14.4.2 Decomposition of regions.
• 14.4.3 An algorithm for the domain decomposition method.
• 14.4.4 The method of domain augmentation to a rectangle.
• 15 Methods for Solving Elliptic Equationsin Curvilinear Orthogonal Coordinates.
• 15.1 Posing boundary-value problems for differential equations.
• 15.1.1 Elliptic equations in a cylindrical system of coordinates.
• 15.1.2 Boundary-value problems for equations in a cylindrical coordinate system.
• 15.2 The solution of difference problems in cylindrical coordinates.
• 15.2.1 Difference schemes without mixed derivatives in the axially-symmetric case.
• 15.2.2 Direct methods.
• 15.2.3 The alternating-directions method.
• 15.2.4 The solution of equations defined on the surface of a cylinder.
• 15.3 Solution of difference problems in polar coordinate systems.
• 15.3.1 Difference schemes for equations in a circle or a ring.
• 15.3.2 The solubility of the boundary-value difference problems.
• 15.3.3 The superposition principle for a problem in a circle.
• 15.3.4 Direct methods for solving equations in a circle or a ring.
• 15.3.5 The alternating-directions method.
• 15.3.6 Solution of difference problems in a ring sector.
• 15.3.7 The general variable-coefficients case.
• Appendices.
• Construction of the minimax polynomial.
• Translator’s note.