Analysis of the Crouzeix-Raviart Surface Finite Element Method for vector-valued Laplacians

Abstract

Recently, a nonconforming surface finite element was developed to discretize 3d vector-valued compressible flow problems arising in climate modeling. In this contribution we derive an error analysis for this approach on a vector-valued Laplace problem, which is an important operator for fluid-equations on the surface. In our setup, the problem is approximated via edge-integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. The latter is continuous at the midpoints of the edges in each vector component. This setup is numerically efficient and straightforward to implement. For this Crouzeix-Raviart discretization we introduce interpolation estimates, derive optimal error bounds in the H1-norm and L2-norm and present an estimate for the geometric error. Numerical experiments validate the theoretical results.