A Modified Rotated- Q_1 Finite Element for the Stokes Equations on Quadrilateral and Hexahedral Meshes

We construct a modified rotated- Q_1 finite element, where we replace the P_2 bubbles of the Rannacher–Turek rotated- Q_1 element by multi-piece linear polynomials. In 2D, we use {1,x,y,|λ _1|} as the basis on a general quadrilateral, where λ _1 is a linear polynomial which vanishes at the middle edge x_3x_4 of two opposite mid-edge nodes x_1 and x_2 . In 3D, we use {1,x,y,z, |λ _1|,|λ _3|} as the basis on a general hexahedron, where λ _1 is a linear polynomial which vanishes at the mid plane x_3x_4x_5x_6 between the two opposite mid-face nodes x_1 and x_2 , and λ _3 is a linear polynomial which vanishes at the mid plane x_1x_2x_5x_6 between the two opposite mid-face nodes x_3 and x_4 . The new rotated- Q_1 finite element is shown inf-sup stable and quasi-optimal in solving the Stokes equations, on general quadrilateral and hexahedral meshes. Numerical tests in 2D and 3D show the method does converge quasi-optimally on non-asymptotic parallelogram or parallelepiped meshes, while the Rannacher–Turek rotated- Q_1 element fails to converge on such meshes.