A lowest order divergence-free finite element on rectangular grids

Abstract

It is shown that the conforming $Q_{2,1;1,2^{-}}{Q}_1^{\prime }$ mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, $Q_{2,1;1,2}=Q_{2,1}\times Q_{1,2}$, and Q_2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. $Q_1^{\prime }$ is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, $Q_1^{\prime }$ is the divergence of the discrete velocity space Q_2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the Raviart-Thomas element.

Keywords

Mixed finite element / Stokes / divergence-free element / quadrilateral element / rectangular grid

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