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Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

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Abstract

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.

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Authors and Affiliations

  1. Institute for Numerical Simulation (INS), University of Bonn, Wegelerstr. 6-8, 53115, Bonn, Germany

    Peter Oswald

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  1. Peter Oswald
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Correspondence to Peter Oswald.

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The work was triggered by a question by C. Carstensen after a talk given by the author at the 2016 European Finite Element Fair about the results from [14].

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Oswald, P. Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error. Appl Math 62, 433–457 (2017). https://doi.org/10.21136/AM.2017.0150-17

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  • DOI: https://doi.org/10.21136/AM.2017.0150-17

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