Parallel Solution of Boundary Value Problems using OpenMP Technology

Abstract

The paper introduces the results of the theoretical and experimental analysis of the Robust Multigrid Technique (RMT), a computational algorithm designed for the numerical solution of (initial-)boundary value problems for the equations of mathematical physics in black-box software. The purpose of the study was to develop robust, efficient and large-scale granulated algorithm for solving a wide class of nonlinear applied problems. The paper describes the algebraic and geometric parallelisms of the RMT and the multigrid cycle for solving nonlinear problems. The OpenMP technology was used to implement the parallel RMT. Computational experiments related to the solution of the Dirichlet problem for the Poisson equation in the unit cube were performed on a personal computer using 3, 9 and 27 threads (p = 3, 9, 27) and on a multiprocessor computer system with shared memory using 27 threads (p = 27). The highest achieved efficiency of the parallel RMT is E ≈ 0.95 at N > 106 and p = 3 and E ≈ 0.80 at N > 107 and p = 27. Findings of the research reveal that the determining factor affecting the efficiency of the parallel RMT is the limited memory performance of multicore computing systems. The complexity of the sequential iteration of the V-cycle and the parallel iteration of the RMT was theoretically analyzed. The study shows that the parallel iteration of the RMT, implemented on 27 threads, will be executed several times faster than the sequential iteration of the V-loop