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

TL;DR

This paper demonstrates that a specific divergence-free finite element method for the Stokes equations exhibits superconvergence, achieving an order of convergence of k+1 for both velocity and pressure, surpassing the standard optimal order.

Contribution

The paper proves superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element, showing it converges at order $k+1$ for velocity and pressure, with numerical validation.

Findings

Superconvergence order of $k+1$ for velocity and pressure.

Numerical tests confirm theoretical predictions.

Enhanced accuracy over standard convergence rates.

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, 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.