Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

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...

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Bibliographic Details
Published in:Numerische Mathematik 2020-03, Vol.144 (3), p.527-551
Main Author: Guo, Hailong
Format: Article
Language:English
Subjects:
Differential geometry
Discretization
Finite element method
Flat surfaces
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Nonlinear programming
Numerical Analysis
Numerical and Computational Physics
Partial differential equations
Recovery
Simulation
Theoretical
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Summary: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.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-019-01099-7