Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

Barker, Andrew T.; Brenner, Susanne C.; Park, E. H.; Sung, Li-yeng

doi:10.1007/s10915-010-9419-5

Cited by 22 publications

(18 citation statements)

References 22 publications

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“…In recent years, considerable attention has been devoted to the development of efficient iterative solvers for the solution of the linear system of equations arising from the discontinuous Galerkin finite element (DGFEM) discretization of a range of model problems. In the framework of two level preconditioners, scalable non-overlapping Schwarz methods have been proposed and analyzed for the h-version of the DGFEM in the articles [21,18,4,5,8,15,9]. More recently, in [6,11], it has been proved that the non-overlapping Schwarz preconditioners can also be successfully employed to reduce the condition number of the stiffness matrices arising from a wide class of high-order DGFEM discretizations of elliptic problems; see, also, [7].…”

Section: Use Policymentioning

confidence: 99%

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

2013

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(2014) 'Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains.', Journal of scientic computing., 60 (1). pp. 203-227. Further information on publisher's website:http://dx.doi.org/10.1007/s10915-013-9792-y Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s10915-013-9792-y.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. In this article we consider the application of Schwarz-type domain decomposition preconditioners for discontinuous Galerkin finite element approximations of elliptic partial differential equations posed on complicated domains, which are characterized by small details in the computational domain or microstructures. In this setting, it is necessary to define a suitable coarse-level solver, in order to guarantee the scalability of the preconditioner under mesh refinement. To this end, we exploit recent ideas developed in the so-called composite finite element framework, which allows for the definition of finite element methods on general meshes consisting of agglomerated elements. Numerical experiments highlighting the practical performance of the proposed preconditioner are presented.Key words. Composite finite element methods, discontinuous Galerkin methods, domain decomposition, Schwarz preconditioners 1. Introduction. In recent years, considerable attention has been devoted to the development of efficient iterative solvers for the solution of the linear system of equations arising from the discontinuous Galerkin finite element (DGFEM) discretization of a range of model problems. In the framework of two level preconditioners, scalable non-overlapping Schwarz methods have been proposed and analyzed for the h-version of the DGFEM in the articles [21,18,4,5,8,15,9]. More recently, in [6,11], it has been proved that the non-overlapping Schwarz preconditioners can also be successfully employed to reduce the condition number of the stiffness matrices arising from a wide class of high-order DGFEM discretizations of elliptic problems; see, also, [7]. We stress that Schwarz-type preconditioners are particularly suited to DGFEMs, in the sense that uniform scalability of the underlying iterative method may be established without the need to overlap the subdomain partition of the computational mesh. This is a particularly attractive property, since the absence of overlapping subdomains reduces communication between processors on parallel machines. By (uniform) scal...

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confidence: 99%

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

2013

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“…The design of efficient solvers for DG discretizations has been pursued only in the last ten years; and, while classical approaches have been successfully extended to second order elliptic problems, the discontinuous nature of the underlying finite element spaces has motivated the creation of new techniques to develop solvers. Additive Schwarz methods (of overlapping and non-overlapping type) are considered and analyzed in [39,34,2,3,4,11]. Multigrid methods are studied in [41,20,18,17,50,29].…”

Section: Introductionmentioning

confidence: 99%

Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients

et al. 2013

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Abstract. We introduce and analyze two-level and multi-level preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents extra difficulties in the analysis which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and nearly-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. The paper includes an Appendix with a collection of proofs of several technical results required for the analysis.

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“…In the framework of two level preconditioners, scalable non-overlapping Schwarz methods have been proposed and analyzed for the h-version of the DG method in the articles [1,2,6,7,9]. Recently, in [3] it has been proved that the non-overlapping Schwarz preconditioners can also be successfully employed to reduce the condition number of the stiffness matrices arising from a wide class of high-order DG discretizations of elliptic problems.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning High–Order Discontinuous Galerkin Discretizations of Elliptic Problems

Antonietti

Houston

2013

Lecture Notes in Computational Science and Engineering

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Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

Cited by 22 publications

References 22 publications

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients

Preconditioning High–Order Discontinuous Galerkin Discretizations of Elliptic Problems

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