A Family of $Q_{k+1,k}\timesQ_{k,k+1}$ Divergence-Free Finite Elements on Rectangular Grids

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It is shown that the conforming $Q_{k+1,k}\times Q_{k,k+1}$-$Q_k^-$ mixed element is stable, and provides optimal order of approximation for the Stokes equations, on rectangular grids, for all $k\ge2$. Here $Q_{k+1,k}$ stands for the space of continuous piecewise-polynomials of degree $(k+1)$ or less in x direction and of degree k or less in y direction. $Q_k^-$ is the space of discontinuous polynomials of separated degree k or less, with spurious modes filtered. To be precise, $Q_k^-$ is the divergence of the discrete velocity space $Q_{k+1,k}\times Q_{k,k+1}$. Polynomials of different degrees in x and y are used for different components of velocity so that the resulting finite element solution is also divergence-free point wise. This is the first divergence-free element found on nontriangular grids. An efficient iterative method is presented where the discrete pressure function is produced as a byproduct. Numerical tests are provided, confirming the theory.