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Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients and \(H^{1+{r}}\), \({r}>0\), Regularity

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Abstract

In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a generic constant, by the best approximation error in the discrete trial space, and robustness means that the generic constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell’s equations.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX, 77843, USA

    Alexandre Ern & Jean-Luc Guermond

  2. CERMICS (ENPC), 77455 Marne-la-Vallee Cedex 2, France and INRIA Paris, 2 rue Simone Iff, 75589, Paris, France

    Alexandre Ern

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Correspondence to Jean-Luc Guermond.

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This material is based upon work supported in part by the National Science Foundation Grants DMS-1619892, DMS-1620058, by the Air Force Office of Scientific Research, USAF, under Grant/Contract Number FA9550-18-1-0397, and by the Army Research Office under Grant/Contract Number W911NF-15-1-0517. This article was communicated by Doug Arnold and is dedicated to the memory of Christine Bernardi.

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Ern, A., Guermond, JL. Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients and \(H^{1+{r}}\), \({r}>0\), Regularity. Found Comput Math 22, 1273–1308 (2022). https://doi.org/10.1007/s10208-021-09527-7

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  • DOI: https://doi.org/10.1007/s10208-021-09527-7

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