A note on a lowest order divergence‐free Stokes element on quadrilaterals

Abstract

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P1 spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q1‐P0 element is also stable. Although this method is a subspace method of Qin‐Zhang's P1‐P0 element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.