Parallel methods and technologies of domain decomposition

Abstract

Parallel domain decomposition methods for solving 3-D grid boundary value problems, which are obtained by finite-element or finite-volume approximations are considered. These problems present the bottle neck between different stages of mathematical modelling, because the modern requirements to accuracy of grid algorithms provide the necessity of solving the systems of linear algebraic equations with the hundred millions of degrees of freedom and with super-high condition numbers which demand the extremal computing resourses. Multi-parameter versions of algorithms with various domain decomposition dimensions — one-dimensional, two-dimensional and three-dimensional, — with or without overlapping of subdomains and with different kinds of internal conjecture conditions on the adjacent boundaries (Dirichlet, Neuman and Robin). The iterative Krylov processes in the trace spaces are investigated for the different preconditioning approaches: Poincare – Steklov operators, block Cimmino method, alternating Schwartz algorithm of additive type, as well as coarse grid correction which is, in a sense, the simplified version of algebraic multigrid method. The comparative analysis of the criteria of parallelezation for the multiprocessor computer systems is made.