Exact Non-Reflecting Boundary Conditions on Perturbed Domains and <Emphasis Type="Italic">hp</Emphasis>-Finite Elements |
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Authors: | Jr" target="_blank">Tommy L BinfordJr David P Nicholls Nilima Nigam T Warburton |
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Affiliation: | (1) Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA;(2) Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA;(3) Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada |
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Abstract: | For exterior scattering problems one of the chief difficulties arises from the unbounded nature of the problem domain. Inhomogeneous
obstacles may require a volumetric discretization, such as the Finite Element Method (FEM), and for this approach to be feasible
the exterior domain must be truncated and an appropriate condition enforced at the far, artificial, boundary. An exact, non-reflecting
boundary condition can be stated using the classical DtN-FE method if the Artificial Boundary’s shape is quite specific: circular
or elliptical. Recently, this approach has been generalized to permit quite general Artificial Boundaries which are shaped
as perturbations of a circle resulting in the “Enhanced DtN-FE” method. In this paper we extend this method to a two-dimensional
FEM featuring high-order polynomials in order to realize a high rate of convergence. This is more involved than simply specifying
high-order test and trial functions as now the scatterer shape and Artificial Boundary must be faithfully represented. This
entails boundary elements which conform (to high order) to the true boundary shapes. As we show, this can be accomplished
and we realize an arbitrary order FEM without spurious reflections. |
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Keywords: | Non-reflecting boundary conditions hp-finite elements Acoustic scattering Dirichlet-to-Neumann maps Geometric perturbation methods |
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