New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

Alouges, François; Averseng, Martin

doi:10.1007/s00211-021-01189-5

Cited by 6 publications

(4 citation statements)

References 44 publications

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“…Recall that the theory of standard pseudodifferential operators on a smooth surface Γ can be viewed as a generalisation of Fourier analysis on the circle. The use of pseudodifferential properties in both the analysis and numerical analysis of boundary integral equations is both well established, see, e.g., [38,128,39,37,123,91,119,92,3,74], and current [40,17,67,6,2,4,64,23].…”

Section: Discussion Of the Ideas Behind The Proof Of Theorem 21mentioning

confidence: 99%

“…Obtaining the multi-dimensional analogues of these results for a range of different FEMs (including discontinuous Galerkin and interior penalty methods) and different ways of truncating the infinite domain Ω + (absorbing boundary conditions, perfectly-matched layers), remains a very active research area; see [104,120,50,107,108,47,106,133,136,45,28,13,29,134,93,30,60,66,84,135,85,87,61]. 2 There has been much research on designing FEMs that mitigate against the pollution effect; we now briefly discuss research on this in four directions.…”

Section: The Pollution Effect For Finite-element Methods Is Well Unde...mentioning

confidence: 99%

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Does the Helmholtz boundary element method suffer from the pollution effect?

Galkowski¹,

Spence²

2022

Preprint

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In d dimensions, approximating an arbitrary function oscillating with frequency k requires ∼ k d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k and in d dimensions) suffers from the pollution effect if, as k → ∞, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than k d for domain-based formulations, such as finite element methods, and k d−1 for boundary-based formulations, such as boundary element methods).It is well known that the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, and research over the last ∼ 30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with k (and how this depends on both p and properties of the scatterer).In contrast to the h-FEM, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite-element-type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the h-BEM must grow with k fall short of proving this.In this paper, we prove that the h-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays). While the proof of this result relies on using information about the large-k behaviour of Helmholtz solution operators, we show in the case when the obstacle is a 2-d ball how this result can be proved using only Fourier series and asymptotics of Hankel and Bessel functions.

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Section: Discussion Of the Ideas Behind The Proof Of Theorem 21mentioning

confidence: 99%

Section: The Pollution Effect For Finite-element Methods Is Well Unde...mentioning

confidence: 99%

Does the Helmholtz boundary element method suffer from the pollution effect?

Galkowski¹,

Spence²

2022

Preprint

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“…It is shown in [1] that A D admits a self-adjoint extension on the weighted L 2 space L 2 1/ω := u ∈ L 1 loc (D) ∥u∥ 2 1/ω :=…”

Section: Introductionmentioning

confidence: 99%

“…Having at hand a cheap way to evaluate this norm on a manifold with boundary may have important applications. As an example, this is the basis of the preconditioning method for the weakly singular integral equation on D proposed in [1].…”

Section: Introductionmentioning

confidence: 99%

Stability of a weighted L2 projection in a weighted Sobolev norm

Averseng¹

2023

Comptes Rendus. Mathématique

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We prove the stability of a weighted L 2 projection operator onto piecewise linear finite elements spaces in a weighted Sobolev norm. Namely, we consider the orthogonal projections π N ,ω from L 2 (D, 1/ω(x)dx) to X N , where D ⊂ R 2 is the unit disk, ω(x) = 1 − |x| 2 and the spaces (X N ) N ∈N consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of D. We show that π N ,ω is stable in a weighted Sobolev norm, and prove an upper bound on the stability constant that does not depend on N . The result also holds when the disk D is replaced by a more general surface Γ ⊂ R 3 , replacing the weight ω by ω Γ (x) := d(x, ∂Γ), the square root of the distance from x to the manifold boundary of Γ.Résumé. On démontre la stabilité dans une norme de Sobolev à poids, de la projection orthogonale par rapport au produit scalaire d'un espace L 2 à poids, sur une famille d'éléments finis linéaires par morceaux. Plus précisément, soit π N ,ω , de L 2 (D, 1/ω(x)dx) dans X N , où D ⊂ R 2 est le disque unité, ω(x) = 1 − |x| 2 et les espaces (X N ) N ∈N sont des espaces de fonctions continues et linéaires par morceaux sur une famille de triangulations régulière de D. On montre que π N ,ω est stable dans une norme de Sobolev à poids, avec une borne supérieure sur la constante de stabilité qui ne dépend pas de N . Le résultat s'étend au cas de surfaces plus générales Γ ⊂ R 3 , en remplaçant le poids ω par ω Γ (x) := d(x, ∂Γ), la racine carrée de la distance de x à ∂Γ, le bord de Γ.

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