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

Abstract

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz singleand double-layer boundary-integral operators as mappings from L 2 (∂Ω) → H 1 (∂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 L 2 (∂Ω), of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit L 2 (∂Ω) → H 1 (∂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, Spence, arXiv 1608.01035]. Definition 1.1 (Smooth hypersurface)We say that Γ ⊂ R d is a smooth hypersurface if there exists Γ a compact embedded smooth d − 1 dimensional submanifold of R d , possibly with boundary, such that Γ is an open subset of Γ, with Γ strictly away from ∂ Γ, and the boundary of Γ can be written as a disjoint unionwhere each Y ℓ is an open, relatively compact, smooth embedded manifold of dimension d − 2 in Γ, Γ lies locally on one side of Y ℓ , and Σ is closed set with d − 2 measure 0 and Σ ⊂ n l=1 Y l . We then refer to the manifold Γ as an extension of Γ.For example, when d = 3, the interior of a 2-d polygon is a smooth hypersurface, with Y i the edges and Σ the set of corner points.Definition 1.2 (Curved) We say a smooth hypersurface is curved if there is a choice of normal so that the second fundamental form of the hypersurface is everywhere positive definite.Recall that the principal curvatures are the eigenvalues of the matrix of the second fundamental form in an orthonormal basis of the tangent space, and thus "curved" is equivalent to the principal curvatures being everywhere strictly positive (or everywhere strictly negative, depending on the choice of the normal). Definition 1.3 (Piecewise smooth) We say that a hypersurfaceDefinition 1.4 (Piecewise curved) We say that a piecewise smooth hypersurface Γ is piecewise curved if Γ is as in Definition 1.3 and each Γ j is curved.The main results of this paper are contained in the following theorem. We use the notation that a b if there exists a C > 0, independent of k, such that a ≤ Cb. Theorem 1.5 (Bounds on S k L 2 (∂Ω)→H 1 (∂Ω) , D k L 2 (∂Ω)→H 1 (∂Ω) , D ′ k L 2 (∂Ω)→H 1 (∂Ω) ) Let Ω be a bounded Lipschitz open set such that the open complement Ω + := R d \ Ω is connected. (a) If ∂Ω is a piecewise smooth hypersurface (in the sense of Definition 1.3), then, given k 0 > 1, S k L 2 (∂Ω)→H 1 (∂Ω) k 1/2 log k, (1.4)for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved (in the sense of Definition 1.4), then, given k 0 > 1, the following stronger estimate holds for all k ≥ k 0 S k L 2 (∂Ω)→H 1 (∂Ω) k 1/3 log k.(1.5) (b) If ∂Ω is a piecewise smooth, C 2,α hypersurface, for some α > 0, then, given ...

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

“…The results of the present paper arise from a fifth direction, namely using estimates on the restriction of quasimodes of the Laplacian to hypersurfaces from [79,17,77,48,26,78] to prove sharp k-explicit bounds on S k , D k and D k as operators from L 2 (∂Ω) to H 1 (∂Ω). We state these sharp k-explicit bounds in §2 below, and they are proved in the companion paper [39]. In the present paper, we use these new results, in conjunction with the results in Points 3 and 4 above, to obtain answers to Q1 and Q2.…”

“…2 Recap of the L 2 (∂Ω) → H 1 (∂Ω) bounds from [39] The following result was proved in [39, Theorem 2.10]. In stating this result, we use the weighted…”

“…We use the new, sharp bounds on the first two of these norms from [39], quoted here as Theorem 2.1, and the sharp bounds on the third of these norms from [ [44]: the mesh thresholds for quasi-optimality in Theorem 1.10 are sharper than the corresponding ones in [44], and the results are valid for a wider class of obstacles. This sharpening is due to the new, sharp bounds on L 2 (∂Ω) → H 1 (∂Ω) norms of S k , D k , and D k from [39], and the widening of the class of obstacles is due to the bound on (A k,η ) −1 L 2 (∂Ω)→L 2 (∂Ω) for nontrapping obstacles from [9,Theorem 1.13].…”

“…We use the new, sharp bounds on the first two of these norms from [39], quoted here as Theorem 2.1, and the sharp bounds on the third of these norms from [ [44]: the mesh thresholds for quasi-optimality in Theorem 1.10 are sharper than the corresponding ones in [44], and the results are valid for a wider class of obstacles. This sharpening is due to the new, sharp bounds on L 2 (∂Ω) → H 1 (∂Ω) norms of S k , D k , and D k from [39], and the widening of the class of obstacles is due to the bound on (A k,η ) −1 L 2 (∂Ω)→L 2 (∂Ω) for nontrapping obstacles from [9,Theorem 1.13]. In more detail: Theorem 1.4 of [44] is the analogue of our Theorem 1.10 except that the former is only valid when Ω is star-shaped with respect to a ball and C 2,α and the mesh threshold is hk (d+1)/2 ≤ C. Comparing this result to Theorem 1.10 we see that we've sharpened the threshold in the d = 3 case, expanded the class of obstacles to nontrapping ones, and added the additional results (b) and (c).…”

Dispersion analysis of the gradient weighted finite element method for acoustic problems in one, two, and three dimensions

“…Marburg and Schneider investigated the influence of element types on numerical error for acoustic boundary elements. Galkowski et al carried out the wavenumber‐explicit analysis for the Helmholtz h‐BEM and proved that how the number of GMRES iterations must grow with the wavenumber k. You et al investigated the dispersion effect of the meshfree point interpolation method for two‐dimensional acoustic problems. Although these numerical improvements are very promising in reducing the pollution error, they still suffer from the ill‐conditioned system matrices or the lack of efficient iterative solver.…”