This dissertation investigates non-overlapping Domain Decomposition (DD) methods for the Finite Element (FE) solution of the time-harmonic Maxwell's equations. Domain decomposition methods provide effective, efficient preconditioned iterative solution algorithms and are attractive due to their inherent parallel nature.
Non-overlapping DD methods divide the original problem domain into smaller disjoint sub-domains in which local sub-problems are to be solved. Adjacent sub-domains are coupled via transmission conditions (TCs) and an iterative process is used to obtain the solution of the original global problem. The TCs determine in large part the performance of the iterative algorithm.
Here, we first introduce a new DD method, the Interior Penalty- (IP-) DD method, intended for problems exhibiting little or no periodicity. The IP-DD method does away with the auxiliary variables required in previous DD methods and is closely linked to IP Discontinuous Galerkin methods. The method is intended to replace conventional FE solvers by providing superior robustness at reduced computational costs. A unique derivation of a general formulation is given and the method is shown to be robust in solving large practical examples.
However, the IP-DD method possesses two deficiencies: the method exhibits poor iterative solver convergence when the mesh is refined and is non-conservative even for conformal meshes. We remedy these deficiencies by introducing a new DD algorithm that uses a vector auxiliary variable to enforce the TC. To further improve the algorithm's robustness, a new TC with a second order transverse derivative term is introduced. The TC is shown to improve the eigenvalue distribution of the DD matrix by shifting eigenvalues corresponding to transverse electric (TE) evanescent modes away from the origin. The new TC, referred to as the second order TE TC, is shown to provide superior convergence over the conventional complex Robin TC. The new DD method exhibits good scalability with respect to mesh size and is conservative when conformal meshes are employed.
While the DD method with second order TE TC considerably improves upon the IP-DD, it is only effective in preconditioning one set of problematic eigenvalues. The eigenmodes neglected by the TE TC, the transverse magnetic (TM) evanescent modes, present the last impediment to truly robust solver convergence. We address these modes by introducing a full second order TC that includes an additional term with a second order transverse derivative. A scalar auxiliary variable is used to implement a DD method with the new TC. We show theoretically and via numerical example that the full second order TC shifts both TE and TM evanescent eigenvalues away from the origin. Numerical examples demonstrate superior iterative solver convergence and improved scalability with respect to mesh size.