Finite element approximation over multiple coordinate systems

Present numerical analysis research devotes much attention to extending existing methods to curved or complicated geometries, domains which often cannot be described non-singularly by one coordinate system. Also, much analysis of partial differential equations from physics and other sciences breaks down because of the singularities of the coordinate system used. To overcome these difficulties, this work details the construction of finite element triangulations of surfaces over multiple coordinate systems and the approximation of functions on these surfaces. Upon surveying existing methods for triangulation such surfaces, this work develops a theoretical framework for measuring the error in these triangulations and the functions built upon them. This framework permits a rigorous understanding of the convergence properties of these often ‘ad hoc’ methods.