Analysis of a P₁ RT₀ finite element method for linear elasticity with Dirichlet and mixed boundary conditions

In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi-Raugel-likeH(div)-conformingmethod proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal.42(2022) 3711-3734]. Therein, the lowest-orderH(div)-conforming Raviart-Thomasspace (RT0) was added to the classical conformingP(1)xP(0)pair to meet the inf-supcondition, while preserving the divergence constraint and some important featuresof conforming methods. Due to the inf-sup stability of the P-1 circle plus RT(0)xP0pair, alocking-free elasticity discretization with respect to the Lam & eacute; constant lambda can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneousdisplacement boundary problem, that is, the discreteH(1)-norm of the displacement isO(lambda-1)when the external body force is a gradient field. We also consider the mixeddisplacement and stress boundary problem, whoseP(1)circle plus RT(0)discretization shouldbe carefully designed due to a consistency error arising from theRT0part. We pro-pose both symmetric and nonsymmetric schemes to approximate the mixed boundarycase. The optimal error estimates are derived for the energy norm and/orL(2)-norm.Numerical experiments demonstrate the accuracy and robustness of our schemes