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A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

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Abstract

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests.

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Acknowledgments

Part of this work was developed during the visit of the second author at the Pennsylvania State University. Special thanks go to the Center for Computational Mathematics and Applications (CCMA) at the Mathematics Department, Penn State for the hospitality and support. The work of the fourth author was supported in part by NSF DMS-1418843, and NSF DMS-1522615. The work of the first author was partially supported by SIR Project No. RBSI14VT0S “PolyPDEs: Non-conforming polyhedral finite element methods for the approximation of partial differential equations” funded by MIUR.

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Authors and Affiliations

  1. MOX-Laboratory for Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Paola F. Antonietti, Marco Sarti & Marco Verani

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Ludmil T. Zikatanov

  3. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str, bl. 8, 1113, Sofia, Bulgaria

    Ludmil T. Zikatanov

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  1. Paola F. Antonietti
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  2. Marco Sarti
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  3. Marco Verani
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  4. Ludmil T. Zikatanov
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Correspondence to Paola F. Antonietti.

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Antonietti, P.F., Sarti, M., Verani, M. et al. A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems. J Sci Comput 70, 608–630 (2017). https://doi.org/10.1007/s10915-016-0259-9

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