Surface Crouzeix-Raviart element for the Laplace-Beltrami equation
This work provides a nonconforming finite element method for geometric PDEs on surfaces, which is incremental for researchers in numerical analysis and computational geometry.
The paper constructs a surface Crouzeix-Raviart element for the Laplace-Beltrami equation on linear approximated surfaces, establishing optimal error estimates and introducing a superconvergent gradient recovery method. Numerical examples validate the theoretical results and demonstrate superconvergence.
This paper is concerned with the nonconforming finite element discretization of geometric partial differential equations. In specific, we construct a surface Crouzeix-Raviart element on the linear approximated surface, analogous to a flat surface. The optimal error estimations are established even though the presentation of the geometric error. By taking the intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix-Raviart element using only the information of discretization surface. The potential of serving as an asymptotically exact {\it 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.