| We have explored solving finite element linear systems, which are not necessarily positive definite, by direct and iterative methods.;For the direct methods we have assumed that no pivoting is necessary, and showed that the factorization operation counts are highly dependent on the reordering scheme used. We looked in detail at various object topologies, problem sizes and values of p, of the finite element method. We were able to seed an expert system-type algorithm to assist in choosing a reordering scheme for the general problem, and to estimate the factorization operation counts for some general 2-D problems.;We then investigated the stopping criterion for the well-known conjugate gradient type methods. We analyzed in detail the required number of iterations for the one dimensional, positive definite elliptic problem, as a function of the prescribed tolerance and a parametrized family of starting values. The analysis was performed without any preconditioning.;We developed the error minimization, in terms of the sequence of Krylov vectors, and derived precise estimates of the required number of iterations for three starting values. In two cases the iteration count was independent of the size of the problem, for large enough problems, while for the third we regained the conventional dependence on the problem size. Therefore, we were not able to derive overall conclusions regarding the preference of the direct vs. iterative solver.;Finally, we described the finite element formulation for the Helmholz equation with radiation boundary conditions in one and two dimensions, for simple rectangular objects. We showed how the straightforward direct method suffers from pivoting issues. While keeping the sampling rate constant, the state-of-the-art iterative method exhibited poor convergence properties for large values of the frequency parameter.;We introduced a new approach which is primarily a direct method plus iterations of quick triangular solves. Initial results indicate that the number of iterations is low, exhibiting this method as a viable alternative for these types of problems. |
