Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

For exterior scattering problems one of the chief difficulties arises from the unboundednature of the problem domain. Inhomogeneous obstacles may require a volumetricdiscretization, such as the Finite Element Method (FEM), and for this approach to be feasiblethe exterior domain must be truncated and an appropriate condition enforced at thefar, artificial, boundary. An exact, non-reflecting boundary condition can be stated usingthe classical DtN-FE method if the Artificial Boundary’s shape is quite specific: circularor elliptical. Recently, this approach has been generalized to permit quite general ArtificialBoundaries which are shaped as perturbations of a circle resulting in the “EnhancedDtN-FE” method. In this paper we extend this method to a two-dimensional FEM featuringhigh-order polynomials in order to realize a high rate of convergence. This is more involvedthan simply specifying high-order test and trial functions as now the scatterer shape andArtificial Boundary must be faithfully represented. This entails boundary elements whichconform (to high order) to the true boundary shapes. As we show, this can be accomplishedand we realize an arbitrary order FEM without spurious reflections.