Comparison of two methods for the computation of singular solutions in elliptic problems

Abstract

We compare two numerical methods for the solution of elliptic problems with boundary singularities. The first is the integrated singular basis function method (ISBFM), a finite-element method in which the solution is approximated by standard polynomial basis functions supplemented by the leading terms of the local (singular) solution expansion. A double application of Green's theorem reduces all Galerkin integrals containing singular terms to boundary integrals with nonsingular integrands. The originally essential boundary conditions are weakly enforced by means of Lagrange multipliers. The second method is a singular function boundary integral method which can be viewed as a modification of the ISBFM. The solution is approximated only by the leading terms of the local solution expansion. The discretized equations are boundary integrals and the dimension of the problem is reduced by one. The two methods are applied to the cracked-beam problem giving very accurate estimates of the leading singular coefficients. Comparisons are made and their limitations are discussed.