Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

Abstract

By the standard theory, the stable Q _k+1,k−Q _k,k+1/$Q_{k}^{dc'}$ divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order k for the velocity in H ¹-norm and the pressure in L ²-norm. This is due to one polynomial degree less in y direction for the first component of velocity, which is a Q _k+1,k polynomial of x and y. In this manuscript, we will show by supercloseness of the divergence free element that the order of convergence is truly k+1, for both velocity and pressure. For special solutions (if the interpolation is also divergence-free), a two-order supercloseness is shown to exist. Numerical tests are provided confirming the accuracy of the theory.

Keywords

Mixed finite element / Stokes equations / Divergence-free element / Quadrilateral element / Rectangular grids / Supercloseness / Superconvergence