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A lowest order divergence-free finite element on rectangular grids
Yunqing HUANG, Shangyou ZHANG
Front Math Chin. 2011, 6 (2): 253-270.
https://doi.org/10.1007/s11464-011-0094-0
It is shown that the conforming Q2,1;1,2-Q1′ mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q2,1;1,2=Q2,1×Q1,2, and Q2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q1′ is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q1′ is the divergence of the discrete velocity space Q2,1;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.
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