Variable reordering for a parallel envelope method

Abstract

We use a parallel direct solver based on the Schur complement method for solving large sparse linear systems arising from the finite element (FE) method. A FE mesh is decomposed into the submeshes by a domain decomposition method. The submatrices formed from the submeshes consist of internal and boundary variables. The internal variables are factorized by an envelope method. Prior to the solution, the variables of each submatrix are reordered to minimize the size of its envelope. The boundary variables are ordered last. The Sloan algorithm is used to perform the reordering, but it does not distinguish between internal and boundary variables. We discuss issues of reordering variables in submatrices and introduce a modified version of the Sloan algorithm that takes the boundary variables into consideration . Experiments on 2 sets of benchmarks show that submatrices produced by the proposed modified Sloan algorithm have profiles smaller by approximately 15%. The new reordering algorithm improved the time of solving FE problems with a parallel envelope solver using a standard domain decomposition method by 23%.