MULTI-P PRECONDITIONERS

A class of algebraic multi-p methods for solving the p-version of the finite element equations is first presented and discussed. It is then shown how these multi-p methods can be used as preconditioners for the conjugate gradient (CG) method. In particular, it is shown that given any preconditioner M-p to CG, a multi-p preconditioner B-p based on M -p can be constructed, which leads to a smaller condition number (and hence faster convergence). Numerical experiments on representative pro blems indicate that the condition numbers after multi-p preconditionin gs are, in fact, independent of p. The numerical results also show gre ater efficiency for the preconditioned conjugate gradient (PCG) method with the multi-p preconditioners in terms of number of iterations and CPU time when compared with two sophisticated linear equation solvers : (1) a direct frontal solver specially designed for the p-version of the finite element analysis; (2) a highly tuned PCG code in ITPACKSC, http://www.netlib.org.itpack/, 1994. Preliminary comparisons of the nu mber of iterations are also made with ROCKITS [Solvers International, Inc., Boulder, CO, 1994], a new commercial code used for the p-version .