Efficient use of singular hierarchical vector functions for the FEM solution of multiscale problems

A number of new hierarchical and singular (scalar and vector) basis functions for field representation in two- and three-dimensional cells have been recently developed and discussed in a series of papers, and in a new book. The hierarchical functions are designed for adaptive p-refinement on domains meshed with differently shaped cells. The polynomial order of the functions may vary from cell to cell to facilitate application to multiscale problems, and to permit one to use large cells. Additive singular basis function sets are needed to efficiently handle corner and wedge singularities. In this paper we take advantage of the fact that (1) each singular basis function can be computed on the fly and (2) that the FEM integrals involving these functions can also be analytically determined provided that the singular cells are rectilinear. It is here proven that this results in (1) a noticeable reduction of the computation time since there is no need to use sophisticated integration routines, (2) an augmented accuracy of the numerical results, and (3) the possibility to include in the model a very high number of singularity coefficients. For sake of brevity the reported results consider the two-dimensional test case of the L-shaped waveguide.