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.

翻译：迹有限元方法已成为求解曲面偏微分方程的热门选择，尤其在处理曲面与体效应耦合的问题中。此类方法通过将偏微分方程定义的连续曲面与三维（体）四面体网格近似相交，形成曲面网格。在经典的$H^1$相容迹方法中，曲面有限元空间通过将体有限元空间限制在曲面网格上得到。然而，如何类似地获得其他重要类型的有限元空间（如$H({\rm div})$相容空间），目前尚不明确。基于Olshanskii、Reusken和Xu在$H^1$相容方法方面的前期工作，本文针对Laplace-Beltrami问题发展了一种"拟迹"混合方法。有限元网格采用曲面与规则四面体体网格的相交结果（如前所述），由此生成的曲面三角剖分高度非结构化且各向异性，但满足经典最大角条件。在此网格上应用混合方法。本文证明了关于体网格尺寸的误差最优估计，以及标量误差投影和后处理标量近似量的超收敛估计。