Effective parallelization is needed to enable the computational analysis of systems that are controlled by coupled, nonlinear, three-dimensional phenomena. The modeling of melt crystal growth systems poses special challenges arising from the need to couple three-dimensional heat and mass transfer, incompressible flow, and a phase-change interface whose position and shape is unknown, a priori. The most robust steady-state formulation of the phase-change problem, termed the isotherm method, tracks the location of the melt-crystal interface via a constraint-like formulation that forces the interface to follow the melting point isotherm. This formulation is implicit in terms of the interface unknowns; however, when these equations are solved using Newton's method and a direct solution algorithm, quadratic convergence yields an efficient algorithm. This approach has proven to be very effective for the solution of two-dimensional problems.

However, the Jacobian matrix arising from the isotherm method is non-symmetric and indefinite, causing significant problems for the convergence of iterative linear solvers that are required for effective parallelization of the three-dimensional problem. While decoupling the field equations for transport from the interface constraints has produced some success for the solution of these problems, the resulting algorithm exhibits only linear convergence. Solution of the fully coupled, simultaneous solution of field and interface unknowns would allow for faster convergence and more cost-effective algorithms.

We propose a new approach to solving such systems by partitioning the Jacobian matrix used in the Newton's method via a Schur complement approach. By partitioning the interface equations from the field equations, this approach maps the difficult part of the problem away to be solved separately and then to be fed back to the rest of the solution. We verify the success of this approach via a test problem based on a two-dimensional crystal growth problem, using a preconditioned GMRES method to solve the block problems. We compare the convergence and expense of this method to that using a direct solver. Prospects for solution of the three-dimensional problem will be demonstrated.