A lowest order divergence-free finite element on rectangular grids

It is shown that the conforming Q _2,1;1,2-Q′₁ 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 × 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 is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q′₁ 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.