The method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities

Abstract

Babuska and Oh have introduced a new approach called the method of auxiliary mapping (MAM), to deal with elliptic boundary value problems with singularities. They showed that for the Laplace equation with corner singularities, in the context of the p-version of the finite element method, MAM yielded an exponential rate of convergence at virtually no extra cost. In this paper those results are extended by showing the exponential convergence of MAM for homogeneous Laplace equations with boundary singularities and Helmholtz equations with both corner and boundary singularities. In addition a convergence result is developed for MAM as applied to the h-p version of the finite element method. To clarify the power of MAM a series of benchmark runs are made for three examples using our implementation, MAPFEM. Comparisons are made with two h-version finite element codes (PLTMG6 and FESOP), both of which use adaptive meshes, the finite difference code ELLPACK, and the recent singular element code ISBFM. The examples include the well-known Motz problem and both homogeneous and nonhomogeneous Helmholtz equations.