As well known the solution of linear elliptic 2nd order PDEs describing heat transfer and elasticity problems, can be expressed by an asymptotic series in the vicinity of singular points in 2-D domains or edges/verticies in 3-D domains. It is typically characterized by a sequence of eigenpairs and their corresponding coefficients. The determination of these eigenpairs, and reliable computation of the coefficients of the asymptotic expansion will be addressed in this talk. These are of practical engineering importance because failure theories involve them.

Several methods for the computation of eigen-pairs (modified Steklov method, the determinant method, and others), and the determination of the coefficients of the asymptotic expansion for 2-D singularities (space enrichment, dual-singular integral also known as CIM, and complementary energy method) will be presented and discussed in terms of robustness, efficiency and ease of generalization. Some of the methods are implemented in a p-FEM code, and their performance will be illustrated (superconvergent rates of the computed data is demonstrated).

In 3-D domains, the singular asymptotic expansion is more complicated: it contains edge singularities, vertex singularities and vertex-edge singularities. The characteristics of vertex singularities can be computed as a straight forward extension of the methods encountered in 2-D domains, whereas edge-singularities have a special structure including so-called shade eigen-functions. The special representation of the edge singularities for the simplified Laplace operator will be presented, and methods for the computation of their characteristics will be demonstrated by numerical methods. Special methods for the computation of the edge flux intensity functions, developed by M. Costabel and M. Dauge are used in conjunction with the p-FEM and numerical examples are shown.

*On Sabbatical leave at: Division of Applied Mathematics, Brown University, Providence, RI, USA