Stability and approximability of the 𝒫1–𝒫0 element for Stokes equations

In this paper we study the stability and approximability of the ??₁–??₀ element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for Stokes equations. Although this element is unstable for all meshes, it provides optimal approximations for the velocity and the pressure in many cases. We establish a relation between the stabilities of the ??₁–??₀ element (bilinear/constant on quadrilaterals) and the ??₁–??₀ element. We apply many stability results on the ??₁–??₀ element to the analysis of the ??₁–??₀ element. We prove that the element has the optimal order of approximations for the velocity and the pressure on a variety of mesh families. As a byproduct, we also obtain a basis of divergence‐free piecewise linear functions on a mesh family on squares. Numerical tests are provided to support the theory and to show the efficiency of the newly discovered, truly divergence‐free, ??₁ finite element spaces in computation. Copyright © 2006 John Wiley & Sons, Ltd.