A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming P1) elements and the pressure by piecewise constants. This method is stable for general meshes (without minimal or maximal angle condition). Denoting by Ne the number of elements in the mesh, the interpolation and consistency errors are of the optimal order h ∼ N−1/3 e which is proved for tensor product meshes. As a by-product, we analyse also nonconforming prismatic elements with P1 ⊕ span {x2 3 } as the local space for the velocity where x3 is the direction of the edge.