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

We construct a modified rotated- \(Q_1\) $Q_1$ finite element, where we replace the \(P_2\) $P_2$ bubbles of the Rannacher–Turek rotated- \(Q_1\) $Q_1$ element by multi-piece linear polynomials. In 2D, we use \(\{1,x,y,|\lambda _1|\}\) $\{1,x,y,|{\lambda }_1|\}$ as the basis on a general quadrilateral, where \(\lambda _1\) ${\lambda }_1$ is a linear polynomial which vanishes at the middle edge \(\textbf{x}_3\textbf{x}_4\) ${\mathbf{x}}_3{\mathbf{x}}_4$ of two opposite mid-edge nodes \(\textbf{x}_1\) ${\mathbf{x}}_1$ and \(\textbf{x}_2\) ${\mathbf{x}}_2$ . In 3D, we use \(\{1,x,y,z, |\lambda _1|,|\lambda _3|\}\) $\{1,x,y,z,|{\lambda }_1|,|{\lambda }_3|\}$ as the basis on a general hexahedron, where \(\lambda _1\) ${\lambda }_1$ is a linear polynomial which vanishes at the mid plane \(\textbf{x}_3\textbf{x}_4\textbf{x}_5\textbf{x}_6\) ${\mathbf{x}}_3{\mathbf{x}}_4{\mathbf{x}}_5{\mathbf{x}}_6$ between the two opposite mid-face nodes \(\textbf{x}_1\) ${\mathbf{x}}_1$ and \(\textbf{x}_2\) ${\mathbf{x}}_2$ , and \(\lambda _3\) ${\lambda }_3$ is a linear polynomial which vanishes at the mid plane \(\textbf{x}_1\textbf{x}_2\textbf{x}_5\textbf{x}_6\) ${\mathbf{x}}_1{\mathbf{x}}_2{\mathbf{x}}_5{\mathbf{x}}_6$ between the two opposite mid-face nodes \(\textbf{x}_3\) ${\mathbf{x}}_3$ and \(\textbf{x}_4\) ${\mathbf{x}}_4$ . The new rotated- \(Q_1\) $Q_1$ 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- \(Q_1\) $Q_1$ element fails to converge on such meshes.