Vector‐Valued Nonconforming Finite Element for Surface Flows

ABSTRACT In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In this contribution, we derive an error analysis for a vector‐valued Crouzeix–Raviart element. The derivation is performed on the vector‐valued Laplace problem, which includes the symmetrical strain rate tensor, an important operator for modeling flow problems. The approximation of the strain rate tensor with the Crouzeix–Raviart element leads to oscillations in the velocity field due to a nontrivial kernel. We derive a stabilized form of the equation and present optimal error bounds in the ‐norm for the Crouzeix–Raviart finite element. The theoretical findings are supported by numerical results.