## 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.

##### Department

##### Description

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