Comparison of linear system solvers applied to diffusion-type finite element equations

Various iterative methods for solving the linear systems associated with finite element approximations to self-adjoint elliptic differential operators are compared based on their performance on serial and parallel machines. The methods studied are all preconditioned conjugate gradient methods, differing only in the choice of preconditioner. The preconditioners considered arise from diagonal scaling, incomplete Cholesky decomposition, hierarchical basis functions, and a Neumann-Dirichlet domain decomposition technique. The hierarchical basis function idea is shown to be especially effective on both serial and parallel architectures.