The Singular Function Boundary Integral Method for Elliptic Problems with Boundary Singularities

Abstract

We review the Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution. The unknowns to be calculated are the singular coefficients, i.e. the coefficients of the local asymptotic expansion, also called generalized stress intensity factors. The discretized Galerkin equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the values of which are additional unknowns. In the case of two-dimensional Laplacian problems, we have shown that this method converges exponentially with respect to the number of singular functions. This is demonstrated via several benchmark applications, including ones involving the biharmonic operator which can be viewed as an extension of the theory.