The Stefan Problem Solved Via Conjugate Gradient—Like Iterative Methods On a Parallel Vector Machine

Abstract

The aim of this paper is to illustrate the validity and efficiency of iterative methods for solving large linear systems arising from the finite element discretizations of the equation governing conduction-controlled solidi fication processes. Starting from the basic enthalpy equation, two alternative formulations are obtained and fixed-grid finite element discretizations are devel oped. These discretizations yield a set of nonlinear equations that are linearized using the Newton-Raph son scheme. The linearized equations are used as a basis for evaluating different iterative methods of the conjugate gradient type. Symmetric scaling and in complete factorization preconditioning of the linear equations are used to improve the convergence prop erties of the iterative methods. Vectorization and paral lelization are also employed to make full use of the CRAY-2 supercomputer. The results indicate that the implementation of currently available iterative solvers leads to efficient solution methodologies for phase change problems.