Optimized Schwarz Domain Decomposition Methods for Scalar and Vector Helmholtz Equations

Antoine, Xavier; Geuzaine, Christophe

doi:10.1007/978-3-319-28832-1_8

Cited by 5 publications

(7 citation statements)

References 42 publications

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“…In the truncated problem defined on Ω, these two operators are localized using a Padé-type approximation of the square root, and the restriction of the obtained approximate boundary conditions on Γ x and Γ y are used. It leads to equations (8) and (9) and the approximate solution u(x, y).…”

Section: Compatibility Relations In Two Dimensionsmentioning

confidence: 99%

“…Unfortunately, because of the highly oscillatory nature of the wave fields, these solvers lead to discretizations with a large number of unknowns and require the solution of large poorly-conditioned linear systems. Research on accurate and computationally efficient methods is very active: we can mention for instance recent works on high-frequency boundary element methods [7,21,22], high-order finite element methods [14,16,57,68] and domain decomposition methods [9,10,34,77].…”

Section: Introductionmentioning

confidence: 99%

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Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

Modave

Geuzaine

Antoine

2020

Journal of Computational Physics

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This paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of highfrequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two-and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle.

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Section: Compatibility Relations In Two Dimensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

Modave

Geuzaine

Antoine

2020

Journal of Computational Physics

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“…point of adjacency of three interfaces (or one interface meeting the boundary of the compuational domain), see Figure 1 above. In another series of contributions Antoine, Geuzaine and their collaborators [2,3,32,33,55] considered the case of impedance coefficients approaching appropriate Dirichlet-to-Neumann maps and obtained fastly converging numerical methods. Here also, the numerical methods were observed to be of good quality only when the subdomain partition does not contain any cross point.…”

Section: Introductionmentioning

confidence: 99%

Non-local variant of the optimised Schwarz method for arbitrary non-overlapping subdomain partitions

Claeys

2021

ESAIM: M2AN

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We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form "identity + non-expansive".

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“…The problem (2) primarily considered in the present manuscript does not a priori lend itself to boundary integral equation techniques simply because (2) is a problem of propagation in heterogeneous media i.e. the PDEs involve a priori varying coefficients.…”

Section: Potential Theorymentioning

confidence: 99%

“…1 above. In another series of contributions Antoine, Geuzaine and their collaborators [2,25,24,5,39] considered the case of impedance coefficients approaching appropriate Dirichlet-to-Neumann maps and obtained fastly converging numerical methods. Here also, the numerical methods were observed to be of good quality only when the subdomain partition does not contain any cross-point.…”

Section: Introductionmentioning

confidence: 99%

A new variant of the Optimised Schwarz Method for arbitrary non-overlapping subdomain partitions

Claeys

2019

Preprint

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We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that can accomodate the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form "identity + contraction".

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Optimized Schwarz Domain Decomposition Methods for Scalar and Vector Helmholtz Equations

Cited by 5 publications

References 42 publications

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

Non-local variant of the optimised Schwarz method for arbitrary non-overlapping subdomain partitions

A new variant of the Optimised Schwarz Method for arbitrary non-overlapping subdomain partitions

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