NUMERICAL ESTIMATION OF COERCIVITY CONSTANTS FOR BOUNDARY INTEGRAL OPERATORS IN ACOUSTIC SCATTERING

Coercivity is an important concept for proving existence and uniqueness of solutions to variational problems in Hilbert spaces. But while coercivity estimates are well known for many variational problems arising from partial differential equations, they are still an open problem in the context of boundary integral operators arising from acoustic scattering problems, where rigorous coercivity results have so far only been established for combined integral operators on the unit circle and sphere. The fact that coercivity holds, even in these special cases, is perhaps surprising, as formulations of Helmholtz problems are generally thought to be indefinite. The main motivation for investigating coercivity in this context is that it has the potential to give error estimates for the Galerkin method which are both explicit in the wavenumber k and valid regardless of the approximation space used; thus they apply to hybrid asymptotic-numerical methods recently developed for the high frequency case. One way to interpret coercivity is by considering the numerical range of the operator. The numerical range is a well established tool in spectral theory, and algorithms exist to approximate the numerical range of finite dimensional matrices. We can, therefore, use Galerkin projections of the boundary integral operators to approximate the numerical range of the original operator. We prove convergence estimates for the numerical range of Galerkin projections of a general bounded linear operator on a Hilbert space to justify this approach. By computing the numerical range of the combined integral operator in acoustic scattering for several interesting convex, nonconvex, smooth, and polygonal domains, we numerically study coercivity estimates for varying wavenumbers. We find that coercivity holds, uniformly in the wavenumber k, for a wide variety of domains. Finally, we consider a trapping domain, for which there exist resonances (also called scattering poles) very close to the real line, to demonstrate that coercivity for a certain wavenumber k seems to be strongly dependent on the distance to the nearest resonance

Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering - extended version

Understanding the spectral properties of boundary integral operators in acoustic scattering has important practical implications, such as for the analysis of the stability of boundary element discretisations or the convergence of iterative solvers as the wavenumber k grows. Yet little is known about spectral decompo- sitions of the standard boundary integral operators in acoustic scattering. Theoretical results are mainly available on the unit disk, where these operators diagonalise in a simple Fourier basis. In this paper we investigate spectral decompositions for more general smooth domains. Based on the decomposition of the acoustic Green’s function in elliptic coordinates we give spectral decompositions on ellipses. For general smooth domains we show that approximate spectral decompositions can be given in terms of circle Fourier modes transplanted onto the boundary of the domain. An important underlying question is whether or not the operators are normal. Based on previous numerical investigations it appears that the standard boundary integral operators are normal only when the domain is a ball and here we prove that this is indeed the case for the acoustic single layer potential. We show that the acoustic single, double and conjugate double layer potential are normal in a scaled inner product on the ellipse. On more general smooth domains the operators can be split into a normal component plus a smooth perturbation. Numerical computations of pseudospectra are presented to demonstrate the nonnonnormal behaviour on general domains

Runge-Kutta convolution coercivity and its use for time-dependent boundary integral equations

A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretisations. It is known that this coercivity property is inherited by convolution quadrature time discretisation based on A-stable multistep methods, which are of order at most two. Here we study the ques- tion as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge– Kutta methods and hence for methods of arbitrary order. As an illustration, the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretisation of a non-linear boundary integral equation that originates from a non-linear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretisation

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators

Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the h-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k→∞ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how h must decrease with k to maintain k-independent quasi-optimality of the Galerkin solutions as k→∞ when the obstacle is nontrapping

Numerical analysis for electromagnetic scattering from nonlinear boundary conditions

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic field along the boundary. Based on time-dependent jump conditions of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretising the nonlinear system of boundary integral equations with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates