# Integration of hp-adaptivity with a two grid solver: applications to electromagnetics

## Abstract

James Clerk Maxwell published in A Treatise on Electricity and Magnetism (1873) a set of partial differential equations governing the electromagnetic (EM) phenomena. Since then, a variety of numerical methods have raised attempting to solve this set of equations for particular applications. Among these numerical methods, a Finite Element (FE) self adaptive hp-refinement algorithm has recently (2001) been developed in the Institute for Computational Engineering and Sciences (ICES), at The University of Texas at Austin. The algorithm produces automatically a sequence of optimal hpmeshes that deliver exponential convergence rates with respect to the number of unknowns (for problems with and without singularities), which allows for high accuracy approximations of solutions to Maxwell’s equations for a large number of EM applications. The fully automatic hp-adaptive strategy iterates along the following steps. First, given an arbitrary (coarse) hp-mesh, the mesh is refined globally in both h and p to yield a fine mesh (h/2, p + 1). The fine mesh solution is used then to approximate the coarse mesh error function, which is utilized to guide optimal hp-refinements of the coarse grid. Critical to the success of the adaptive strategy is the solution of the fine grid problem, which may increase the problem size (with respect to the coarse grid problem) at least by one order of magnitude. In this dissertation, we have studied and implemented an efficient two grid solver algorithm (coupled with the hp-adaptive strategy) that is suitable for a large class of problems in 2D and 3D, including elliptic and electromagnetic boundary value problems, for both real and complex valued operators, with applications to Radar Cross Section (RCS) analysis, modeling of Logging While Drilling EM measuring devices, and design of waveguides. We have first implemented the classical V-cycle algorithm for symmetric and positive definite problems by combining an overlapping Block-Jacobi smoother with a relaxation parameter, and a direct solve on the coarse grid. For EM problems, an extra Block-Jacobi smoother designed to control gradients has been implemented, as proposed by Hiptmair. The convergence rate for the resulting two grid solver is independent of the mesh size h (provided that the coarse grid is fine enough), and depends only logarithmically on the order of approximation p. A theoretical convergence analysis has been illustrated with extensive numerical experimentation in 2D and 3D. These results show a significant convergence improvement when the relaxation parameter is selected to be optimal. The numerical experiments also prove that, using a two grid solver, it is possible to guide the optimal hp-refinements with only a partially converged fine grid solution. Finally, several electromagnetic applications illustrate the efficiency of combining the hp-adaptivity with the two grid solver. Three particular applications have been studied in this dissertation: model problems relevant to Radar Cross Section (RCS) analysis, modeling of Logging While Drilling EM measuring devices, and design of waveguides.