Analysis of a P_1⊕RT_0 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-like H(div) -conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowest-order H(div) -conforming Raviart–Thomas space ( RT_0 ) was added to the classical conforming P_1× P_0 pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of the P_1⊕RT_0× P_0 pair, a locking-free elasticity discretization with respect to the Lamé constant λ can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete H^1 -norm of the displacement is 𝒪(λ ^-1) when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose P_1⊕RT_0 discretization should be carefully designed due to a consistency error arising from the RT_0 part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or L^2 -norm. Numerical experiments demonstrate the accuracy and robustness of our schemes.