A Two-Dimensional Self-Adaptive $hp$ Finite Element Method for the Analysis of Open Region Problems in Electromagnetics

A two-dimensional self-adaptive $hp$ finite element method (FEM) for the analysis of open region problems in electromagnetics is described. The method incorporates an iterative procedure for solving open region problems. More precisely, solution of the open problem is achieved by solving a low number of closed domain problems with the same matrix, so the computational cost for the second and subsequent problems is very small. Thus, the particularities due to the open nature of the problem are hidden and, among other advantages, self-adaptive strategies developed for conventional closed domains and, specifically, of $hp$ type, can be used without modifications. The $hp$-FEM discretization is made in terms of quadrangles/triangles of variable order of approximation supporting anisotropy and hanging nodes. The adaptive strategy is fully automatic and is based on minimizing the interpolation error (by using the projection of the error from a fine grid) delivering exponential convergence rates for the energy error, even in the presence of singularities.