Higher Order Finite Elements and Adaptivity in Computational Electromagnetics

Abstract: This thesis is concerned with efficient and accurate numerical methods for the solution of Maxwell's equations. In particular, the finite element method with higher order elements and adaptive mesh-refinement is considered.

The main contribution of the thesis is a new method for computing goal-oriented error estimates for the scattering parameters of waveguide structures. Expressions for the errors that are based on the solutions to dual problems are derived using the recently developed Dual Weighted Residual method. Then, a new set of hierarchical basis functions with certain orthogonality properties is developed in order to facilitate the efficient computation of approximate dual solutions. The approximate dual solutions are in turn used to obtain approximate values of the errors for of the scattering parameters.

Numerical studies of both two and three-dimensional problems are used to demonstrate the performance of the error estimates. Numerical studies also show that a very accurate and efficient scheme is obtained by the combination of higher order elements and adaptive mesh-refinement, guided by the proposed error estimates.

A part of the thesis is also concerned with the finite-element/finite-difference time-domain hybrid method, particularly in combination with adaptive mesh-refinement.