Error analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems

Authors:
Leonid Chindelevitch;David P. Nicholls;Nilima Nigam
Affiliations:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montréal, Que., Canada H3A 2K6;Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607, USA;Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montréal, Que., Canada H3A 2K6
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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Abstract

In this paper we present an error analysis for a high-order accurate combined Dirichlet-to-Neumann (DtN) map/finite element (FE) algorithm for solving two-dimensional exterior scattering problems. We advocate the use of an exact DtN (or Steklov-Poincare) map at an artificial boundary exterior to the scatterer to truncate the unbounded computational region. The advantage of using an exact DtN map is that it provides a transparent condition which does not reflect scattered waves unphysically. Our algorithm allows for the specification of quite general artificial boundaries which are perturbations of a circle. To compute the DtN map on such a geometry we utilize a boundary perturbation method based upon recent theoretical work concerning the analyticity of the DtN map. We also present some preliminary work concerning the preconditioning of the resulting system of linear equations, including numerical experiments.