In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

/pdf/a-galerkin-boundary-element-method-for-high-frequency-1hv7wfzk91.pdf

A Galerkin Boundary Element Method for High Frequency Scattering by Convex Polygons

Discontinuous Galerkin methods with plane waves for time-harmonic problems

The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides a variational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.

/pdf/error-estimates-for-the-ultra-weak-variational-formulation-1g06u9mvao.pdf

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

In comparison with low-order finite element methods (FEMs), the use of oscillatory basis functions has been shown to reduce the computational complexity associated with the numerical approximation of Helmholtz problems at high wave numbers. We compare two different wave element methods for the 2D Helmholtz problems. The methods chosen for this study are the partition of unity FEM (PUFEM) and the ultra-weak variational formulation (UWVF). In both methods, the local approximation of wave field is computed using a set of plane waves for constructing the basis functions. However, the methods are based on different variational formulations; the PUFEM basis also includes a polynomial component, whereas the UWVF basis consists purely of plane waves. As model problems we investigate propagating and evanescent wave modes in a duct with rigid walls and singular eigenmodes in an L-shaped domain. Results show a good performance of both methods for the modes in the duct, but only a satisfactory accuracy was obtained in the case of the singular field. On the other hand, both the methods can suffer from the ill-conditioning of the resulting matrix system.

/pdf/comparison-of-two-wave-element-methods-for-the-helmholtz-1xn548vviv.pdf

Comparison of two wave element methods for the Helmholtz problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is ${\cal O}(N^{-(\nu+1)} \log^{1/2}N)$, where the number of degrees of freedom is proportional to $N\log N$. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

/pdf/a-wavenumber-independent-boundary-element-method-for-an-4gtfvksqnk.pdf

A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

The ultra-weak variational formulation has been used effectively to solve time-harmonic acoustic and electromagnetic wave propagation in inhomogeneous media. We develop the ultra-weak variational formulation for elastic wave propagation in two space dimensions. In order to improve the accuracy and stability of the method, we find it necessary to approximate the S- and P-wave components of the solution in a balanced way. Some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements.

/pdf/the-ultra-weak-variational-formulation-for-elastic-wave-2akdg01fu3.pdf

F. Collino

Papers

The Ultra-Weak Variational Formulation for Elastic Wave Problems