Mesh theorems on traces, normalizations of function traces and their inversion

Nepomnyaschikh, S. V.

doi:10.1515/rnam.1991.6.3.223

Cited by 51 publications

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“…It was shown that nonoverlapping domain decomposition methods converge independently of the contrast (e.g. [17,20] and [27], Sects. 6.4.4 and 10.2.4) when conductivity variations within coarse regions are bounded.…”

Section: Introductionmentioning

confidence: 99%

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

et al. 2012

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Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461-1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.Mathematics Subject Classification. 65F10, 65N20, 65N22, 65N30, 65N55.

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Section: Introductionmentioning

confidence: 99%

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

et al. 2012

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“…As our theoretical convergence results do not depend on !, but are based on extension theorems, such numerical results illustrate the fact that there exists a mesh extension theorem for Domain II in spite of the failure of the continuous extension theorem, but with a mesh-dependent constant Ä h slowly decreasing with the mesh size h, see Reference [21] and references therein.…”

Section: Numerical Resultsmentioning

confidence: 88%

Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

Бахвалов

Knyazev

Parashkevov

2002

Numerical Linear Algebra App

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Dedicated to Olof B. Widlund on the occasion of his 60th birthday. SUMMARYWe prove extension theorems in the norms described by Stokes and LamÃ e operators for the threedimensional case with periodic boundary conditions. For the LamÃ e equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem.Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and ÿctitious domain methods, and in analysis of ÿnite element methods. We consider an application of established extension theorems to an e cient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coe cients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coe cient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning.Under some natural assumptions on smoothness and topological properties of subdomains with small coe cients, we prove convergence of the simplest Richardson method uniform in the jump of coecients. For the LamÃ e equations, the convergence is also uniform in the incompressible limit.Our preliminary numerical results for two-dimensional di usion problems show fast convergence uniform in the jump and in the mesh size parameter.

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“…Not surprisingly, our preconditioner fits neatly into the abstract framework of the fictitious space lemma due to Nepomnyaschikh [29], see also Oswald [33] and Xu [37].…”

Section: A Domain Embedding Methodsmentioning

confidence: 99%

A domain embedding method for Dirichlet problems in arbitrary space dimension

Rieder¹

1998

ESAIM: M2AN

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Mesh theorems on traces, normalizations of function traces and their inversion

Cited by 51 publications

References 0 publications

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

A domain embedding method for Dirichlet problems in arbitrary space dimension

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