On the Applicability of Lions’ Energy Estimates in the Analysis of Discrete Optimized Schwarz Methods with Cross Points

Gander, Martin J.; Kwok, Félix

doi:10.1007/978-3-642-35275-1_56

Cited by 15 publications

(18 citation statements)

References 7 publications

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“…At the discrete level, a condition number estimate for a finite element discretizations of isotropic diffusion problems and Robin transmission conditions can be found in [33]. Two different consistent discretizations at cross points for finite element discretizations were derived and analyzed in [22], and optimized Robin parameters at cross points at the algebraic level were derived in [20], but classical energy estimates can not directly be used in the presence of cross points [21]. There is also to the best of the authors knowledge no study so far on efficient coarse spaces for anisotropic diffusion problems.…”

Section: Discussionmentioning

confidence: 99%

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Gander

Halpern

Hubert

et al. 2020

Moroccan Journal of Pure and Applied Analysis

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We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

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Section: Discussionmentioning

confidence: 99%

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Gander

Halpern

Hubert

et al. 2020

Moroccan Journal of Pure and Applied Analysis

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“…We further assume that L i is a diagonal matrix with support along artificial interfaces only; in addition to simplifying our analysis below, this choice arises naturally as part of the discrete integration by parts formula, cf. [11].…”

Section: This Gives the Definition (4)mentioning

confidence: 99%

“…By symmetry, we assume that the left and right edges are identical in the sense that (11) T S 0 = S 0 T, where…”

Section: 3mentioning

confidence: 99%

“…Optimized Schwarz methods can also be formulated algebraically in the discrete setting [29,27], with properties similar to the continuous case when the mesh is fine enough and no cross points are present, i.e., no grid points are adjacent to three or more subdomains. However, the correct formulation of OSMs in the presence of cross points is a delicate problem, since it is difficult to discretize the PDE and boundary conditions while maintaining continuity of the solution [11]. In [23], Loisel presents a general formulation for OSM and the closely related 2-Lagrange multiplier method (2LM) that handles cross points systematically.…”

Section: Introductionmentioning

confidence: 99%

“…Lions showed in [22], using an energy estimate argument, that for any choice of p ij > 0, (2) converges to the unique solution of (1) in L 2 (Ω i ) and weakly in L 2 (∂Ω i ). However, as noted in [11], the discretization of (2) is delicate when cross points are present, since one must weakly impose two different sets of Robin conditions for each piece of the interface at the cross point. A similar difficulty has been observed in [28] in the optimized Schwarz preconditioner in the context of solving the magnetohydrodynamics equations.…”

Section: Introductionmentioning

confidence: 99%

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Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Gander¹,

Kwok²

2012

SIAM J. Sci. Comput.

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Optimized Schwarz methods are domain decomposition methods in which a largescale PDE problem is solved by subdividing it into smaller subdomain problems, solving the subproblems in parallel, and iterating until one obtains a global solution that is consistent across subdomain boundaries. Fast convergence can be obtained if Robin conditions are used along subdomain boundaries, provided that the Robin parameters p are chosen correctly. In the case of second order elliptic problems such as the Poisson equation, it is well known for two-subdomain problems without overlap that the optimal choice is p = O(h −1/2) (where h is the mesh size), with the resulting method having a convergence factor of ρ = 1 − O(h 1/2). However, when cross points are present, i.e., when several subdomains meet at a single point, this choice leads to a divergent method. In this article, we show for a model problem that convergence can only occur if p = O(h −1) at the cross point; thus, a different scaling of the Robin parameter is needed to ensure convergence. In addition, this choice of p allows us to recover the 1 − O(h 1/2) convergence factor in the resulting method.

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Corners and stable optimized domain decomposition methods for the Helmholtz problem

2021

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On the Applicability of Lions’ Energy Estimates in the Analysis of Discrete Optimized Schwarz Methods with Cross Points

Cited by 15 publications

References 7 publications

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Corners and stable optimized domain decomposition methods for the Helmholtz problem

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