A class of iterative methods for finite element equations

A general method in the form of an accelerated preconditioned iterative refinement method is presented for the solution of symmetric, sparse matrix problems, and some inherently advantageous properties of the conjugate gradient acceleration method are pointed out.

Abstract

A general method in the form of an accelerated preconditioned iterative refinement method (including some wellknown iterative methods and direct factorization methods) is presented for the solution of symmetric, sparse matrix problems. An analysis of one such approximate factorization, the SSOR method, is given, and some inherently advantageous properties of the conjugate gradient acceleration method are pointed out. A comparison is made of the computational complexity and storage in the SSOR preconditioned method with some direct methods applied to second order discretized boundary value problems. For plane problems of average size the direct methods are somewhat faster if enough right hand sides are present. For large enough problems (large number of nodes) the iterative method is faster. For three-dimensional problems no Cholesky factorization method can compete with the SSOR preconditioned method, not even for average sized problems.