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 dc k 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.

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“…In such a method, the discrete velocity solution is also divergence‐free pointwise. We call such finite elements divergence‐free elements, which are studied in [4, 6, 9, 11–13, 15, 16, 18, 20–25].…”

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