Frequency-adapted Galerkin boundary element methods for convex scattering problems

Ecevit, Fatih; Ozen, H. Cagan

doi:10.1007/s00211-016-0800-7

Cited by 18 publications

(73 citation statements)

References 25 publications

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“…Therefore, in applications to acoustic scattering, the invertibility of A k,η on L 2 (Γ) is also important. Indeed, L 2 (Γ) is a natural function space setting for implementation and analysis of Galerkin numerical methods for the solution of the direct equations (6.6) and (6.12), and the indirect equation (6.8) (e.g., [59,22,44,35,38] and recall the discussion in §1.5.3).…”

Section: Integral Equations For the Exterior Dirichlet Problemmentioning

confidence: 99%

“…1, but this bound on the inverse does not imply the stronger (6.31). The advantage of coercivity, as opposed to just boundedness of the inverse, for the numerical analysis of Galerkin methods is discussed in [86]; for example, the coercivity result in [86] completes the numerical analysis of high frequency numerical-asymptotic boundary element methods for scattering by convex obstacles [33,35].…”

Section: Counterexample To a Conjecture On Coercivitymentioning

confidence: 99%

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

Chandler-Wilde¹,

Spence²,

Gibbs³

et al. 2020

SIAM J. Math. Anal.

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This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realised through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss.This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in 2-d), or two cubes (in 3-d), whose sides are parallel. We show, via developments of the vectorfield/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral-equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analysing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle.

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Section: Integral Equations For the Exterior Dirichlet Problemmentioning

confidence: 99%

Section: Counterexample To a Conjecture On Coercivitymentioning

confidence: 99%

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

Chandler-Wilde¹,

Spence²,

Gibbs³

et al. 2020

SIAM J. Math. Anal.

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“…By the no-occlusion condition of [16] and formulated in Remark 1, the periodic orbit is such that rays close to it are amenable to geometrical optics. We assert that a solution to (14) exists in a neighbourhood of the points τ * j on each Γ j , j = 1, . .…”

Section: Remarkmentioning

confidence: 99%

“…In order to find a local solution to (14), we augment the system with the initial conditions (11). Furthermore, it is clear that the phase φ 1 is only determined up to a constant, which we may choose, e.g., by prescribing a value for φ 1 (τ * 1 ).…”

Section: Remarkmentioning

confidence: 99%

“…Furthermore, it is clear that the phase φ 1 is only determined up to a constant, which we may choose, e.g., by prescribing a value for φ 1 (τ * 1 ). In the following section, we will resort to Taylor series expansions of the unknown functions to approximate the local solution of (14). This approach does lead to fully explicit expressions for the coefficients.…”

Section: Remarkmentioning

confidence: 99%

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On the eigenmodes of periodic orbits for multiple scattering problems in 2D

Huybrechs

Opsomer

2019

International Journal of Mechanical Sciences

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Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles converge to an equilibrium after a few iterations. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. The results may be used to accelerate ray tracing schemes after a small number of initial iterations.

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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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Frequency-adapted Galerkin boundary element methods for convex scattering problems

Cited by 18 publications

References 25 publications

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

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

On the eigenmodes of periodic orbits for multiple scattering problems in 2D

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

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