Generalized Finite Element Method for Vector Electromagnetic Problems

The generalized finite element method (GFEM), as introduced by Babuska, is a partition of unity framework for solving scalar partial differential equations (PDEs). To date, they have been applied extensively to the solution of elliptic and parabolic PDEs. This technique can be interpreted as a unified prescription of a host of well known techniques such as classical finite element methods that are based on tessellation, point cloud techniques, and element free Galerkin methods. The focus of this paper is to extend this technique to solve vector electromagnetic problems. However, modification of GFEM to address this problem poses a novel set of challenges: i) the vector nature of the problem and the different continuity requirements on each component imply that basis functions developed should share similar characteristics; ii) the basis functions have to be able to represent divergence free electromagnetic fields (in a source free region) when they themselves are not divergence free. These have been the primary impediments to extension of this technique. In this paper, we propose a method that may be used to overcome these obstacles, and will demonstrate h and p convergence characteristics of the proposed method for a range of problems. Finally, we will demonstrate convergence characteristics of this technique for analyzing scattering from a sharp wedge when the basis functions include analytical "local" field representation