Quadratic Nonconforming Finite Element Method For 3D Stokes Equations

In 2014, Meng et. al. introduced a quadratic nonconforming brick element (MSLK element) to approximate 3D second-order elliptic problems. Shortly thereafter, Bai and Meng constructed a stable mixed finite element scheme to solve Stokes equations, by adding bubble functions onto each element based on the MSLK element with piecewise discontinuous Pi element. This paper presents a new quadratic nonconforming mixed finite element method for 3D Stokes equations based on the MSLK element. We show the instability of a mixed finite element pair where the velocity and pressure are approximated by the vector-valued MSLK element and the piecewise discontinuous P1 element, respectively, and point out that the finite element linear system is of a 4-dimensional rank deficiency. Then we enrich this discrete velocity space by a 4-dimensional global bubble function space such that the enriched pair satisfies a weak stability. This mixed finite element space is a subspace of Bai and Meng’s with least degrees of freedom, and thus re-duces large amounts of computations. Optimal order of convergence is still achieved. Numerical tests verify the theoretical analysis of our new pair.