Adaptive solution strategy for solving large systems ofp-type finite element equations

Abstract

A solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p-type finite element method using a hierarchical basis function set. The algorithm dynamically branches to either direct or iterative solution methods. In. the iterative solution branch, the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient (PCG) method. The convergence rate of the PCG algorithm is monitored to improve the heuristics used in the choice of the preconditioner. The robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations. This strategy has been implemented in a p-version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems.