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

Abstract

SUMMARYIn this paper we study the stability and approximability of the P 1 -P 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 Q 1 -P 0 element (bilinear/constant on quadrilaterals) and the P 1 -P 0 element. We apply many stability results on the Q 1 -P 0 element to the analysis of the P 1 -P 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, P 1 finite element spaces in computation.

A C1-P2 finite element without nodal basis

“…Further it is divergence-free, i.e., div curl u ≡ 0 on Ω. By Theorem 5.1 of [22], curl u is a linear combination of local div-free basis functions shown in Figure 6, i.e.,…”

“…We note that Theorem 2.1 can be shown directly, without using the result [22], by the dimension counting of Strang's conjecture, which shall be discussed in next section.…”

“…Such div-free vectors form a local basis for P 1 -P 0 mixed-finite elements, approximating the velocity and the pressure in Stokes or Navier-Stokes equations, see [22]. This is in fact, how this new C 1 -P 2 finite element was discovered.…”

“…Recently, we found a divergence-free, local basis for continuous P 1 elements on the criss-cross grids ( Fig. 1(C)) in [22], which is originated in [1,21]. Because the C 0 divergence-free piecewise-P 1 vector space is the curl of C 1 piecewise-P 2 space on the same triangulation, in this paper we find the anti-derivatives of the above basis to get a local basis for the C 1 -P 2 space on the criss-cross grid.…”

Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

“…There are several other such divergence-free finite elements, cf. [2,14,16,17,19,20,[29][30][31]33].…”