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

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.

Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods

“…Numerical solutions of the time-harmonic FSI problem have been studied widely using the FEM ( [4-7, 11-13, 28, 32, 39]), or the boundary integral equation (BIE) methods ( [31,38]). Here, we declare that our main concerns consist of the analysis of the DtN boundary condition as it is coupled with FEM to solve problems in unbounded domains, without intentions of performing sophisticated finite element analysis for the model problem ( [5][6][7][11][12][13]), or for time-harmonic waves ( [33]).…”

Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis

“…The present article is designed to make contributions to the error analysis in the latter application. As an extension of the standard DtN-FEM, the authors of [9] developed a new method for the realization of exact non-reflecting boundary conditions without the restriction on the shape of artificial boundary, and carried out a sequence of works [10][11][12] on the numerical analysis and computation. Corresponding to the nonlocal boundary conditions, there are several type of local boundary conditions [13][14][15].…”