Optimal Discrete Transmission Conditions for a Nonoverlapping Domain Decomposition Method for the Helmholtz Equation

Magoulès, Frédéric; Roux, François‐Xavier; Salmon, Stéphanie

doi:10.1137/s1064827502415351

Cited by 55 publications

(36 citation statements)

References 17 publications

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“…In (Magoulès et al, 2004b;Magoulès et al, 2005;Magoulès et al, 2006), it was proposed to approximate the optimal choice of A 1 and A 2 , which is based on the Schur complement of the entire neighboring subdomain, by a Schur complement on patches reaching from the interface Γ into the neighboring subdomain. We use here one patch per subdomain interface, denoted by P 1 and P 2 , with external boundaries Γ 1 and Γ 2 , see Fig.…”

Section: Patch Substructuring Methodsmentioning

confidence: 99%

“…Instead of computing the Schur complement of the entire patch, one can also compute Schur complements of smaller parts of the patch and then add them to approximate the Schur complement of the entire patch, see (Magoulès et al, 2004b;Magoulès et al, 2005;Magoulès et al, 2006). The present analysis does not apply to this case, and this additional approximation requires further studies.…”

Section: Theorem 3 the Classical Schwarz Methods (18)-(19) And The Sumentioning

confidence: 99%

“…If the entire Schur complement is used, optimal iteration numbers can be achieved like at the continuous level with the SteklovPoincaré operator, see (Magoulès et al, 2004b). Approximations are, however, obtained differently, namely, by computing approximate Schur complements based on patches.…”

Section: Mj Gander Et Almentioning

confidence: 99%

“…The following theorem shows the equivalence of an entire class of decomposed problems to the underlying original problem, see (Magoulès et al, 2004b) …”

mentioning

confidence: 99%

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Analysis of Patch Substructuring Methods

Gander¹,

Halpern²,

Magoulès³

et al. 2007

International Journal of Applied Mathematics and Computer Science

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Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

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Section: Patch Substructuring Methodsmentioning

confidence: 99%

Section: Theorem 3 the Classical Schwarz Methods (18)-(19) And The Sumentioning

confidence: 99%

Section: Mj Gander Et Almentioning

confidence: 99%

“…The following theorem shows the equivalence of an entire class of decomposed problems to the underlying original problem, see (Magoulès et al, 2004b) …”

mentioning

confidence: 99%

See 2 more Smart Citations

Analysis of Patch Substructuring Methods

Gander¹,

Halpern²,

Magoulès³

et al. 2007

International Journal of Applied Mathematics and Computer Science

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“…Inspired in part by the earlier work on algebraic Schwarz methods, in this paper, we mimic the philosophy of optimized Schwarz methods when solving block banded linear systems; see also [18], [19], [20]. Our approach consists of optimizing the block which would correspond to the artificial interface (called transmission matrix), so that the spectral radius of the iteration operator is reduced; see section 4.…”

mentioning

confidence: 99%

An Optimal Block Iterative Method and Preconditioner for Banded Matrices with Applications to PDEs on Irregular Domains

Gander¹,

Loisel²,

Szyld³

2012

SIAM J. Matrix Anal. & Appl.

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Abstract. Classical Schwarz methods and preconditioners subdivide the domain of a partial differential equation into subdomains and use Dirichlet transmission conditions at the artificial interfaces. Optimized Schwarz methods use Robin (or higher order) transmission conditions instead, and the Robin parameter can be optimized so that the resulting iterative method has an optimized convergence factor. The usual technique used to find the optimal parameter is Fourier analysis; but this is only applicable to certain regular domains, for example, a rectangle, and with constant coefficients. In this paper, we present a completely algebraic version of the optimized Schwarz method, including an algebraic approach to find the optimal operator or a sparse approximation thereof. This approach allows us to apply this method to any banded or block banded linear system of equations, and in particular to discretizations of partial differential equations in two and three dimensions on irregular domains. With the computable optimal operator, we prove that the optimized Schwarz method converges in no more than two iterations for the case of two subdomains. Similarly, we prove that when we use an optimized Schwarz preconditioner with this optimal parameter, the underlying minimal residual Krylov subspace method (e.g., GMRES) converges in no more than two iterations. Very fast convergence is attained even when the optimal transmission operator is approximated by a sparse matrix. Numerical examples illustrating these results are presented.

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Optimal convergence properties of the FETI domain decomposition method

Maday

Magoulès

2006

Numerical Methods in Fluids

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SUMMARYIn this paper an original variant of the FETI domain decomposition method is introduced for heterogeneous media. This method uses new absorbing interface conditions in place of the Neumann interface conditions defined in the classical FETI method. The optimal convergence properties of the classical FETI method and of its variant are first demonstrated, both in the case of homogeneous and heterogeneous media. Secondly, novel and efficient absorbing interface conditions, which avoid rigid body motions, are investigated and analysed. Numerical experiments illustrate the dependence of the proposed method upon several parameters, and confirm the robustness and efficiency of this method when equipped with such absorbing interface conditions.

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Optimal Discrete Transmission Conditions for a Nonoverlapping Domain Decomposition Method for the Helmholtz Equation

Cited by 55 publications

References 17 publications

Analysis of Patch Substructuring Methods

Analysis of Patch Substructuring Methods

An Optimal Block Iterative Method and Preconditioner for Banded Matrices with Applications to PDEs on Irregular Domains

Optimal convergence properties of the FETI domain decomposition method

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