The generalized finite element method (GFEM), first introduced by Babuska, is a partition of unity-based solver for scalar partial differential equations (PDEs). To date, they have been applied extensively to the solution of elliptic and parabolic PDEs. This technique is a generalization of a host of well known methods for solving PDEs, specially the classical finite element method(FEM), element free galerkin(EFG), hp clouds, etc. The main goal of this dissertation is for developing a similar methodology for vector electromagnetic problems. Developing a solution to these problems necessitates addressing the following problems: (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). (iii) Development of appropriate boundary conditions to truncate the computational domain is necessary. (iv) High condition number of the resulting system also plagues GFEM solver, as it does other high order solvers. Solution to these problems, and the developments of the GFEM solver is presented here for both time and frequency domains. In any case, the h- and p- convergence of the method is presented.
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