Testing Parallel Linear Iterative Solvers for Finite Element Groundwater Flow Problems

Abstract

The modeling of groundwater flow using three-dimensional finite element discretizations creates a need to solve large systems of sparse linear equations (Ax = h) at each of several nonlinear iterations. These linear systems can be difficult to solve because of the ill- conditioning of the matrix A resulting from the widely varying magnitudes of its coefficients. Because the meshes may contain millions of nodes, iterative solvers are typically used to perform the Ax = b solution. Since 80 percent or more of the computational time is spent in the iterative solver part of the computer program, choosing the most efficient solver for each application can dramatically reduce the total solution time. This paper tests 12 Krylov subspace parallel linear iterative solvers with five preconditioners (60 scenarios) on linear systems of equations resulting from a finite element study of remediation of a military site using pump-and-treat technology. Both symmetric, positive-definite matrices, resulting from a Picard linearization of the nonlinear equations for the steady-state case, and nonsymmetric matrices, arising from a Newton linearization of the nonlinear equations of a transient case, are studied. The portable, extensible toolkit for scientific computation (PETSc) library was used in this study on the Engineer Research and Development Center Major Shared Resource Center SGI O3K and Cray XT3 computers. Matrix data corresponding to each subdomain in a parallel groundwater finite element program were first written to a file in a compressed sparse column format. A separate program was then written in FORTRAN to read these data, renumber the nodes, call the PETSc routines to load A andb and then solve for x, and finally compute error norms. Solver time, iteration count, 2-norm and co- norm of the residual after convergence, weak speedup, and strong speedup are tabulated in this paper for the different scenarios and then analyzed.