Edge flux intensity functions in polyhedral domains and their extraction by a quasidual function method

The asymptotics of solutions to scalar second order elliptic boundary value problems in three-dimensional polyhedral domains in the vicinity of an edge is provided in an explicit form. It involves a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the edges. Utilizing the explicit structure of the solution in the vicinity of the edge we present a new method for the extraction of the EFIFs called quasidual function method. It can be interpreted as an extension of the dual function contour integral method in 2-D domains, and involves the computation of a surface integral J[R] along a cylindrical surface of radius R away from the edge as presented in a general framework in (Costabel et al., 2004). The surface integral J[R] utilizes special constructed extraction polynomials together with the dual eigen-functions for extracting EFIFs. This accurate and efficient method provides a polynomial approximation of the EFIF along the edge whose order is adaptively increased so to approximate the exact EFIF. It is implemented as a post-solution operation in conjunction with the p-version finite element method. Numerical realization of some of the anticipated properties of the J[R] are provided, and it is used for extracting EFIFs associated with different scalar elliptic equations in 3-D domains, including domains having edge and vertex singularities. The numerical examples demonstrate the efficiency, robustness and high accuracy of the proposed quasi-dual function method, hence its potential extension to elasticity problems.