Quadratic nonconforming finite element method for 3D Stokes equations on cuboid meshes

Abstract

In this paper, the quadratic nonconforming brick element (MSLK element) introduced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK−P 1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P 1 element approximates the pressure. As a cure, we adopt the piecewise P 1 macroelement to discretize the pressure instead of the standard piecewise P 1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.