Computing singular solutions of elliptic boundary value problems in polyhedral domains using the p-FEM

Numerical methods for computing singular solutions of linear second order elliptic partial differential equations (Laplace and Elasticity problems) in polyhedral domains are presented. The singularities may be caused by edges, vertices, or abrupt changes in material properties or boundary conditions. In the vicinity of the singular lines or points the solution can be represented by an asymptotic series, composed of eigen-pairs and their amplitudes. These are of great interest from the point of view of failure initiation because failure theories directly or indirectly involve them. This paper addresses a general method based on the modified Steklov formulation for computing the eigen-pairs and a dual weak formulation for extracting the amplitudes numerically using the p-version of the finite element method. The methods are post-solution operations on the finite element solution vector and have been shown in a two dimensional setting to be super-convergent.