Boundary Value Problems Numerical Solution on Multiblock Grids

Авторы: Martynenko S.I., Varaksin A.Yu. Опубликовано: 15.02.2021
Опубликовано в выпуске: #1(94)/2021  
DOI: 10.18698/1812-3368-2021-1-18-33

Раздел: Математика | Рубрика: Вычислительная математика  
Ключевые слова: boundary value problems, multigrid methods, multi-block grids

Results of theoretical analysis of the geometric multigrid algorithms convergence are presented for solving the linear boundary value problems on a two-block grid. In this case, initial domain could be represented as a union of intersecting subdomains, in each of them a structured grid could be constructed generating a hierarchy of coarse grids. Multigrid iteration matrix is obtained using the damped nonsymmetric iterative method as a smoother. The multigrid algorithm contains a new problem-dependent component --- correction interpolation between grid blocks. Smoothing property for the damped nonsymmetric iterative method and convergence of the robust multigrid technique are proved. Estimation of the multigrid iteration matrix norm is obtained (sufficient convergence condition). It is shown that the number of multigrid iterations does not depend on either the step or the number of grid blocks, if interpolation of the correction between grid blocks is sufficiently accurate. Results of computational experiments are presented on solving the three-dimensional Dirichlet boundary value problem for the Poisson equation illustrating the theoretical analysis. Results obtained could be easily generalized to multiblock grids. The work is of interest for developers of highly efficient algorithms for solving the (initial-) boundary value problems describing physical and chemical processes in complex geometry domains


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