Development of Generalized Finite Element Method for Vector Electromagnetic Problems

Generalized finite element methods, introduces a partition of unity framework that can encompass a host of finite element based technique for solution to scalar partial differential equations. These include the classical finite elements that are based on tessellation as well as point cloud techniques and the element free Galerkin methods. When generalized FEM (GFEM) has been developed for scalar problems, the generalization of GFEM to address problems in computational electromagnetics poses unique challenges: (i) the vector nature of the problem and the different continuity requirements on each component imply that basis functions developed share similar characteristics; (ii) the basis functions have to be divergence free (approximately) in a source free domain. These two have been the primary impediments to extension of this technique. In this paper, we propose methods that may be used to overcome these. We demonstrate the h and p convergence characteristics of the proposed method for a range of problems