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Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

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Abstract

In this paper, we are concerned with the nonconforming finite element discretization of geometric partial differential equations. We construct a surface Crouzeix–Raviart element on the linear approximated surface, analogous to a flat surface. The optimal convergence theory for the new nonconforming surface finite element method is developed even though the geometric error exists. By taking an intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix–Raviart element using only the information of discretized surface. The potential of serving as an asymptotically exact a posteriori error estimator is also exploited. A series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method.

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  1. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Hailong Guo

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This work was partially supported by Andrew Sisson Fund of the University of Melbourne.

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Guo, H. Surface Crouzeix–Raviart element for the Laplace–Beltrami equation. Numer. Math. 144, 527–551 (2020). https://doi.org/10.1007/s00211-019-01099-7

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