Stable QkQk-Qk−1Qk−1 mixed finite elements with discontinuous pressure

Mixed finite elements play a central role in many important CFD applications involving Stokes solvers and alike. A natural mixed finite element for the Stokes equations is the QkQk-Qk−1Qk−1 element on rectangular grids, by which the velocity is approximated by continuous polynomials of separated degree kk and the pressure is approximated by discontinuous polynomials of separated degree k−1k−1. Such an element is, however, not stable. We propose in this paper three modified QkQk-Qk−1Qk−1 elements with certain element-wise divergence-free property of velocity, where the pressure space is slightly restricted to subspaces in C−1C−1-Qk−1Qk−1, yet the optimal order of approximation is still retained. The stability and approximation analysis for the new elements are presented. Comprehensive numerical experiments are also conducted to confirm the theoretical analysis and to reveal the super-convergence for some of these new elements.