Vectorized dissection on the hypercube

Abstract

Dissection ordering is used with Gaussian elimination on a Hypercube parallel processor with vector hardware to solve matrices arising from finite-difference and finite-element discretizations of 2-D elliptic partial differential equations. These problems can be put into a matrix-vector form, Ax = f, where the matrix A takes the place of the differential operator, x is the solution vector, and f is the source vector. The domain is divided among the nodes with neighboring subdomains sharing a strip called a separator. Each processor is given its own part of the source vector and computes its own part of the stiffness matrix, A. The elimination starts out in parallel; communication is only needed after most of the elimination is finished when the edges need to be eliminated. Back substitution is initially done on the domain edges, and then totally in parallel without communication on each node. The Hypercube code involved was optimized to work with vector hardware. Example problems and timings are given with comparisons to nonvector runs.