Parallel Solution Of Finite Element Equations Using Multi-color SSOR-PCG

As a most mature and effective numerical method，the finite element method hasbeen widely used in engineering practices. With the development of computertechnology, large-scale and more complex numerical problems are becoming morecommon. The traditional sequential finite element method faces the challenge of longcomputing time and large memory. It no longer meets the requirements of thedevelopment of numerical calculations and thus needs the help of parallel computingtechnology. The most important work of the finite element method is to solve a seriesof large sparse symmetric positive definite equations. The SSOR-PCG is an effectivemethod for solving this class of equations. In order to implement the parallel solutionof the finite element equations using the SSOR-PCG, this paper carries out thefollowing studies:(1) In order to increase the parallel performance of the SSOR-PCG, the vertexcoloring algorithm of graphs is used for multicolor ordering the nodes of finiteelement meshes.(2) Three different parallel subroutines of the SSOR-PCG suitable for parallelexecution on distributed, shared, and distributed-shared memory computers are codedwith Fortran MPI, OpenMP and MPI+OpenMP, respectively.(3) According to the characteristics of the forward substitution and backwardsubstitution of the multicolor SSOR preconditioner, choice is made for suitable MPIcommunication routines by comparison.(4) Three solutions for avoiding race conditions under OpenMP and MPI+OpenMP are proposed for the use of problems with different scale and parallelcomputers with different memory capacity.