A mixed quasi-trace surface finite element method for the Laplace-Beltrami problem

TL;DR

This paper introduces a novel mixed quasi-trace finite element method for solving the Laplace-Beltrami problem on surfaces, extending trace methods to $H({\rm div})$-conforming spaces with proven optimal and superconvergent error estimates.

Contribution

It develops a new mixed quasi-trace finite element approach for the Laplace-Beltrami problem, enabling $H({\rm div})$-conforming discretizations on intersected surface meshes.

Findings

Optimal error estimates with respect to bulk mesh size.

Superconvergent estimates for scalar error projection.

Effective handling of highly unstructured and anisotropic surface meshes.

Abstract

Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately intersecting the continuous surface on which the PDE is posed with a three-dimensional (bulk) tetrahedral mesh. In classical $H^1$-conforming trace methods, the surface finite element space is obtained by restricting a bulk finite element space to the surface mesh. It is not clear how to carry out a similar procedure in order to obtain other important types of finite element spaces such as $H({\rm div})$-conforming spaces. Following previous work of Olshanskii, Reusken, and Xu on $H^1$-conforming methods, we develop a ``quasi-trace'' mixed method for the Laplace-Beltrami problem. The finite element mesh is taken to be the intersection of the surface with a regular tetrahedral bulk mesh as previously described, resulting in a surface triangulation that is highly unstructured and anisotropic but satisfies a classical maximum angle condition. The mixed method is then employed on this mesh. Optimal error estimates with respect to the bulk mesh size are proved along with superconvergent estimates for the projection of the scalar error and a postprocessed scalar approximation.