Analysis of Some Finite Elements for the Stokes Problem

Abstract

Abstract. We study some finite elements which are used in the approximation of the Stokes problem, so as to obtain error estimates of optimal order.Résumé. Nous étudions deux éléments finis utilisés pour l'approximation du problème de Stokes et obtenons des estimations d'erreur d'ordre optimal.

Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems

“…We first solve the Stokes eigenvalue problem (2.12) by the lowest order Bernardi-Raugel mixed finite element ( [6], [9] and [14]) and solve the Stokes source problem (3.4) by the Q 2 − P 1 mixed finite element on the rectangular meshes ( [9] and [14]). Now, we introduce the lowest order Bernadi-Raugel mixed finite element V h = {v ∈ (H 1 0 (Ω)) 2 : v| e ∈ Q 12 × Q 21 , ∀ e ∈ T h },…”

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

“…(Ω S ) be the Bernardi-Raugel interpolation operator (see [9,30]), which is linear and bounded with respect to the H 1 (Ω S )-norm. We remark that, given v ∈ H 1 ΓS (Ω S ), there holds…”

“…[4,9,11,24,44]). Note that each one of them is named after the unknown to which it is applied later on.…”

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

“…Fig. 2, by the divergence Q k+1,k − Q k,k+1 element (2.3) and by the rotated Bernardi-Raugel element [5,10,21]:…”

“…Ainsworth and Coggins established [1] the stability and the optimal order of convergence for the Taylor-Hood Q k+1 /Q k element, where the pressure space is continuous too. The Bernardi-Raugel element [5] optimizes the Q k+1 /Q dc k−1 element, when k = 1, by reducing the velocity space to Q 1,2 − Q 2,1 polynomials. Here the first component of velocity in the Bernardi-Raugel element is a polynomial of degree 1 in x direction, but of degree 2 in y direction.…”