Block Iterative Methods For The Numerical Solution Of Partial Differential Equations In Applications

Many problems arisen from science, technology and project may come down to theinitial value and boundary problems of the partial differential equations. Few of theseequations can be given an analytical solution while most are given an approximate valueby numerical methods. The numerical methods for partial differential equationsproblems can be partitioned into two parts: one is the discretization of equations, andthe other is the solution of linear algebraic equations. Because the coefficient matrix oflinear algebraic equations from the discretization of partial differential equations has anall-right block structure, we emphasis on the block iterative methods applied innumerical solution of partial differential equations in the text.In the text, we start from the discretization of partial differential equations,introduce in the common discretization methods for elliptic equation and improve anexponent difference scheme for the two-dimension convection-diffusion equationswhich can be regard as an advanced difference scheme for those equations.We detailed introduce the block basic iterative methods for linear algebraicequations in the third chapter. In the solution of the Difichlet boundary problem ofLaplace equations, we provide an effective method for determining the spectral radiusand the best slack variable of block basic iterative methods. At the same time, wepresent the spectral radius and the best slack variable of basic iterative methods from thenine-point difference scheme.In the forth chapter, we introduce the PE(Pseudo Elimination) methods and theirconvergences, especially the convergences with blocked tridiagonal M-matrix andH-matrix. Based on quadratic PEk method, quadratic EPEk method is advanced and itsconvergences are discussed by the conditions of the symmetric positive definite and thepositive-definable Matrix. At the same time, we provide an experimental conclusion forchoosing the parameter of quadratic PEk methods from compact difference scheme forsolving the Dirichlet boundary problem of Laplace equations.Combined with block ILU preconditioners, Krylov subspace methods now are the widely used methods for solving linear algebraic equations from the dicretzation ofpartial differential equations problems. When we introduce several blockpreconditioners for blocked tridiagonal symmetric M-matrix and nonsymmetricH-matrix, we present four block preconditioners for the blocked five-diagonalsymmetric M-matrix. Numerical experiments are carrying out to check the validity.We put the block basic iterative methods, PE methods and block ILUpreconditioned Krylov subspace methods together and compare with each other bymany numerical experiments. Some conclusions are drawn in the sixth chapter.