High-frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis

Chandler-Wilde, Simon N.; Spence, Euan A.; Gibbs, Andrew; Smyshlyaev, V. P.

doi:10.1137/18m1234916

Cited by 25 publications

(33 citation statements)

References 83 publications

(232 reference statements)

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“…Under the assumptions of Case 1a) or 1b), the cornerstone of the analysis is a stability estimate that we establish hereafter. Similar upper bounds are available in [33,42] for the setting considered here, and we refer the reader to [12,13,51] for more complex geometries. However, these estimates are not as sharp as possible and/or the constant C stab ( Ω, Γ D , x 0 ) is not computable, since they have been derived having a priori error estimation (or simply, stability analysis) in mind.…”

Section: 1mentioning

confidence: 89%

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

2021

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We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for the right-hand side, as previously introduced in the space semi-discrete setting, with a novel, piecewise quartic, globally twice-differentiable time-reconstruction of the fully discrete solution. Our main results show that the proposed estimator is reliable and efficient in a damped energy norm. These properties are illustrated in a series of numerical examples.

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Section: 1mentioning

confidence: 89%

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

2021

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“…The activity has occurred in, broadly speaking, four different directions: 3. The use of bounds on the Helmholtz solution operator (also known as resolvent estimates) due to Vainberg [80] (using the propagation of singularities results of Melrose and Sjöstrand [63]) and Morawetz [66] to prove bounds on both (A k,η ) −1 L 2 (∂Ω)→L 2 (∂Ω) and the inf-sup constant of the domain-based variational formulation [22,73,9,24], and also to analyse preconditioning strategies [40]. 4.…”

Section: )mentioning

confidence: 99%

“…We note that[24, Remark 6.6] gives an example of a nontrapping obstacle for which A k,η is not coercive uniformly in k; therefore, the class of obstacles for which A k,η is coercive, uniformly in k, is a proper subset of the class of nontrapping obstacles.…”

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confidence: 99%

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

2019

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We consider solving the exterior Dirichlet problem for the Helmholtz equation with the hversion 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.

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“…3. The use of bounds on the Helmholtz solution operator (also known as resolvent estimates) due to Vainberg [55] (using the propagation of singularities results of Melrose and Sjöstrand [40]) and Morawetz [45] to prove bounds on both (A ′ k,η ) −1 L 2 (∂Ω)→L 2 (∂Ω) and the inf-sup constant of the domain-based variational formulation [12], [48], [4], [13], and also to analyse preconditioning strategies [27].…”

Section: )mentioning

confidence: 99%

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

Galkowski

Spence

2019

Integr. Equ. Oper. Theory

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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 ...

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High-frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis

Cited by 25 publications

References 83 publications

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

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

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

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