Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element

Details Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element

2011-12-16

arxiv.org/abs/1112.3705v1

Mathematics

Numerical Analysis

Scientific paper

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, a $Q_{k+1,k}$ polynomial. In this manuscript, we will show a superconvergence of the divergence free element that the order of convergence is truly $k+1$, for both velocity and pressure. Numerical tests are provided confirming the sharpness of the theory.

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