The finite element method (FEM) is widely used in development of today's electromagnetic technology as it provides accurate and efficient simulation of electromagnetic fields. Recently, techniques such as the vector generalized finite element methods (VGFEM) have been proposed to increase the accuracy and flexibility of the FEM. VGFEM is a general framework that enables inclusion of a large class of basis functions other than interpolatory polynomials in the solution space, and offers several advantages such as: (i) easy implementation of p-adaptivity, (ii) use of mixed orders of polynomial basis functions or use of different basis functions within a simulation domain, and (iii) inclusion of physics in the solution space. Hybridizing VGFEM with FEM can provide a framework for reaping advantages of both methods; resulting in a simulation tool that is highly efficient and accurate. In this dissertation, we study the following topics: (i) A general methodology is developed to adapt the method to polyhedral meshes, and tested for brick and tetrahedral meshes. (ii) Convergence and error characteristics of the method are extensively studied, especially dispersion/phase error characteristics. (iii) Techniques are developed to extend VGFEM to inhomogeneous problems. (iv) Boundary integrals, absorbing boundary conditions, and perfectly matched layers are formulated within the VGFEM framework. (v) VGFEM is hybridized with classical FEM. (vi) Finally, time domain VGFEM is developed for transient electromagnetic problems.
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