A Vectorized Matrix-Vector Multiply and Overlapping Block Iterative Method

Abstract

An overlapping block iterative method is presented for solving large systems of equations which result from finite element discretizations. This algorithm is very much in the spirit of the element-by-element techniques described by Hughes et al. and by Carey and Jiang; however, it differs from these in that the algorithm here is a block iterative method in the classical sense where the blocks or groups are the nodes in the individual elements. Since any given node may appear in several elements, this technique has been called an “overlapping” block iterative method.

This algorithm is particularly attractive for vector processing. It uses a special vectorized form of a matrix-vector multiply in which all computations are done on an element level. The individual element stiffness matrices were stacked together to create long vectors. It has been implemented on the CYBER 205 and has been compared in terms of execution time and storage requirements against standard equation solvers such as (1) band solver, (2) sparse matrix solver, (3) conjugate gradient, (4) SOR and (5) symmetric SOR with conjugate gradient acceleration.

Comparisons were made on a variety of regular grids which ranged in size from several hundred unknown up to 11,000 unknowns, and on several extremely irregular grids which occur in biomedical applications. The element-by-element overlapping block algorithm and vectorized matrix-vector multiply were found to be very efficient, and it was particularly advantageous for problems with irregular finite element grids.