Wavenumber-Explicit Regularity Estimates on the Acoustic Single- and Double-Layer Operators

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from L2(∂Ω)→H1(∂Ω) (where ∂Ω is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in L2(∂Ω), of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit L2(∂Ω)→H1(∂Ω) bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper (Galkowski, Müller, and Spence in Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem, 2017. arXiv:1608.01035)

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

Eigenvalues of the truncated Helmholtz solution operator under strong trapping

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that (a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and (b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalized minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretizations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [P. Marchand et al., Adv. Comput. Math., to appear])

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L2(Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp, and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is curved. Together, these results give bounds on the condition number of the operator on L2(Γ); this is the first time L2(Γ) condition-number bounds have been proved for this operator for obstacles other than balls

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

Optimal constants in nontrapping resolvent estimates and applications in numerical analysis

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds the outgoing resolvent satisfies ∥ ∥ χ R ( k ) χ ∥ L 2 → L 2 ≤ C k − 1 for k > 1 , but the constant C has been little studied. We show that, for high frequencies, the constant is bounded above by 2 π times the length of the longest generalized bicharacteristic of ∣ ∣ ξ ∣ ∣ 2 g − 1 remaining in the support of χ . We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation

Perfectly-matched-layer truncation is exponentially accurate at high frequency

We consider a wide variety of scattering problems including scattering by Dirichlet, Neumann, and penetrable obstacles. We consider a radial perfectly-matched layer (PML) and show that for any PML width and a steep-enough scaling angle, the PML solution is exponentially close, both in frequency and the tangent of the scaling angle, to the true scattering solution. Moreover, for a fixed scaling angle and large enough PML width, the PML solution is exponentially close to the true scattering solution in both frequency and the PML width. In fact, the exponential bound holds with rate of decay c (omicrontanθ − C)k where omicron is the PML width and θ is the scaling angle. More generally, the results of the paper hold in the framework of black-box scattering under the assumption of an exponential bound on the norm of the cutoff resolvent, thus including problems with strong trapping. These are the first results on the exponential accuracy of PML at high-frequency with non-trivial scatterers

Decompositions of High-Frequency Helmholtz Solutions via Functional Calculus, and Application to the Finite Element Method

Over the last 10 years, results from [J. M. Melenk and S. Sauter, Math. Comp., 79 (2010), pp. 1871–1914], [J. M. Melenk and S. Sauter, SIAM J. Numer. Anal., 49 (2011), pp. 1210–1243], [S. Esterhazy and J. M. Melenk, Numerical Analysis of Multiscale Problems, Springer, New York, 2012, pp. 285–324] and [J. M. Melenk, A. Parsania, and S. Sauter, J. Sci. Comput., 57 (2013), pp. 536–581] decomposing high-frequency Helmholtz solutions into “low-” and “high-” frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer–Sjöstrand functional calculus [B. Helffer and J. Sjöstrand, Schrödinger Operators, Springer, Berlin, 1989, pp. 118–197] this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand and Zworski [J. Amer. Math. Soc., 4 (1991), pp. 729–769] thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the hp-finite-element method (hp-FEM) applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighborhood of the obstacle, and (ii) the h-FEM applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the hp-FEM applied to this problem does not suffer from the pollution effect

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

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 decompositions 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