On Fast Domain Decomposition Solving Procedures for hp-Discretizations of 3-D Elliptic Problems

Korneev, V. G.; Langer, Ulrich; Xanthis, Leonidas S.

doi:10.2478/cmam-2003-0034

Cited by 22 publications

(24 citation statements)

References 17 publications

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“…There is a big literature, in particular high order methods and three dimensional problems are treated in [2,5,7,8,9,18,22,26,27,29,29,34,35,36,38,40]. There is a classical paper on multi-level analysis for h-version DG methods by Gopalakrishnan and Kanschat [17], and a recent one studying higher order methods by Antonietti and Houston [3] showing a polynomial growth of the condition number in p. We will see that the conditioning is significantly improved by hybridization, namely to a poly-logarithmic growth.…”

Section: Introductionmentioning

confidence: 87%

Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes

Schöberl

Lehrenfeld

2013

Advanced Finite Element Methods and Applications

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Hybrid discontinuos Galerkin methods are popular discretization methods in applications from fluid dynamics and many others. Often large scale linear systems arising from elliptic operators have to be solved. We show that standard p-version domain decomposition techniques can be applied, but we have to develop new technical tools to prove poly-logarithmic condition number estimates, in particular on tetrahedral meshes.

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Section: Introductionmentioning

confidence: 87%

Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes

Schöberl

Lehrenfeld

2013

Advanced Finite Element Methods and Applications

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“…It would be interesting to perform a comparison in terms of computational efficiency and robustness with respect to the mesh aspect ratio with other solvers based either on multilevel ideas or fast solvers exploiting the tensor product type of the mesh. As investigated in recent studies [10,32,34], a clever combination of all these techniques could lead to an efficient and robust solver for hp finite element approximations.…”

Section: Perspectivesmentioning

confidence: 99%

“…Thus highly stretched meshes with huge aspect ratios are obtained in practice. Consequently, the condition number of the stiffness matrix severely deteriorates: [32] (see also the references therein). Unfortunately, up to now, no iterative substructuring method has been proven to be efficient when very thin elements and/or subdomains (involving meshes with high aspect ratio) or general non quasiuniform meshes are employed.…”

Section: Introductionmentioning

confidence: 99%

“…Several works on domain decomposition have been proposed for higher order approximations [2,27,31,33,40] and more recently [32] (see also the references therein). Unfortunately, up to now, no iterative substructuring method has been proven to be efficient when very thin elements and/or subdomains (involving meshes with high aspect ratio) or general non quasiuniform meshes are employed.…”

Section: Introductionmentioning

confidence: 99%

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A numerical study on Neumann-Neumann methods forhpapproximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Toselli¹,

Vasseur²

2006

ESAIM: M2AN

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Abstract.In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal. 24 (2004) 123-156]. They confirm that the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. Good results are also obtained for certain singularly perturbed problems. The condition numbers only grow polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes [Pavarino, RAIRO: Modél. Math. Anal. Numér. 31 (1997) 471-493]. This paper follows [Toselli and Vasseur, Comput. Methods Appl. Mech. Engrg. 192 (2003) IntroductionIn recent years hp finite element methods have gained an increasing popularity in both the applied mathematics community and some engineering application fields. The hp finite element method has been first introduced by Gui and Babuška [26] and since then some monographs [30,37,50,54] or part of textbooks ([38], Sect. 8.4), have proposed both theoretical and numerical insights into this topic. These methods are found to be particularly useful when high or extremely high accuracy is needed and when minimal dissipation and dispersion errors in the discrete system are required.Indeed the main reason for the interest in hp finite element methods is that they achieve exponential rates of convergence for both regular and singular solutions [37,50]. In presence of singularities or boundary layers, suitably graded meshes, geometrically refined towards corners, edges and/or faces have to be employed to achieve such an exponential rate of convergence [37,50]. Thus highly stretched meshes with huge aspect ratios are obtained in practice. Consequently, the condition number of the stiffness matrix severely deteriorates: [32] (see also the references therein). Unfortunately, up to now, no iterative substructuring method has been proven to be efficient when very thin elements and/or subdomains (involving meshes with high aspect ratio) or general non quasiuniform meshes are employed. Thus our goal is to fill this gap by proposing a domain decomposition preconditioner that is robust with the mesh aspect ratio and possible large jumps in the coefficients. Its theoretical derivation has been presented in [58]. In this paper the main emphasis is devoted to an extensive numerical study of its performances. These robustness aspects will be carefully summarized and analyzed hereafter for some three-dimensional elliptic model problems of diffusion or reaction-diffusion type. These model problems defined on simple geometries have been chosen here in order to be easily reproducible. Nevertheless some of them...

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“…For the Schur complement preconditioner, we refer to References [4,6,9,10]. The extension operator has been considered in References [11,12].…”

Section: Introductionmentioning

confidence: 99%

Improvements for some condition number estimates for preconditioned system in p‐FEM

Beuchler

Braess

2006

Numerical Linear Algebra App

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SUMMARYIn this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the p-version of the FEM. We analyse several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. Relations between the p-version of the FEM and the h-version are helpful in the interpretations of the results.

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On Fast Domain Decomposition Solving Procedures for hp-Discretizations of 3-D Elliptic Problems

Cited by 22 publications

References 17 publications

Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes

Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes

A numerical study on Neumann-Neumann methods forhpapproximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Improvements for some condition number estimates for preconditioned system in p‐FEM

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