On Adaptive Wavelet-based Methods for the Maxwell Equations

The subject of the thesis is the applicability of recently developed wavelet-based methods for efficient numerical solution of the Maxwell equations, which constitute a model for electromagnetic phenomena. An introduction to wavelet bases is given, followed by an introduction to computational electromagetics. Then, variational formulations for the Maxwell equations are developed, and results of well-posedness are proven. We put particular attention to the existence of singularities in the solution to Maxwells equations, and the danger that they may not be resolved by certain numerical schemes. The formulations belong to the class of Fictitious Domain formulations. Among the advantages of such formulations are that they allow the use of simple, for instance periodic, functions for their discretization, and boundary and other conditions need not be incorporated into the employed basis functions. Finally, being an important issue for the adaptive wavelet-based methods, formulations of the Maxwell equations in an infinitely-dimensional Euclidean space setting are considered, and initial results of well-posedness are stated.