New trends in finite element methods

This chapter addresses both the "old" body of knowledge and the "new" trends of research and practice in FEM as applied to electromagnetics. It presents the general mathematical background and numerical components of FEM and discusses FEM formulations, discretizations, and solution procedures, mostly in the context of the higher order FEM computation. This includes the generation of curvilinear elements for higher order modeling of geometry, implementation of polynomial vector basis functions for higher order modeling of fields within the elements, and Galerkin testing method for discretizing the wave equations. The chapter focuses on the higher order FEM as the most general and versatile approach, where the low -order modeling is naturally included in the higher order FEM paradigm.