Adaptive Finite Element Methods for Div-Curl Problems

Doctoral thesis, 2002

In this thesis, we develop and apply finite element methods to problems of div-curl type, mainly from applications in electromagnetics. In particular, we focus on least-squares formulations for problems with singularities, edge elements in eddy current computations, and the implementation of the finite element method. We introduce discontinuous elements in the least-squares finite element method (LSFEM) and enforce continuity and boundary conditions weakly. For this scheme, we prove stability and optimal a priori error estimates for the div-grad and the div-curl problems posed on nonconvex domains. Numerical studies in three dimensions, confirming the theoretical results, are presented. Moreover, combining LSFEM and a Galerkin formulation, a scheme for wave propagation problems with beneficial dispersion properties is proposed. To efficiently solve eddy current problems, we use tetrahedral and hexahedral edge elements in an ungauged potential formulation combined with adaptive mesh refinement. We also introduce anisotropic mesh refinement to compute th e power loss for a hydrogenerator with laminated materials. A software environment for implementation of finite element methods is developed. It is based on object-oriented programming techniques and combines generality with sufficient efficiency.