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

Yunqing Huang , Shangyou Zhang

Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (2) : 143 -162.

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Communications in Mathematics and Statistics ›› 2013, Vol. 1 ›› Issue (2) : 143 -162. DOI: 10.1007/s40304-013-0012-8
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Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

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Abstract

By the standard theory, the stable Q k+1,kQ 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

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Yunqing Huang,Shangyou Zhang. Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids. Communications in Mathematics and Statistics, 2013, 1(2): 143-162 DOI:10.1007/s40304-013-0012-8

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