In this paper we present an a posteriori error estimator for the stabilized P 1 nonconforming finite element method of the linear elasticity problem based on a nonsymmetric H(div)-conforming approximation of the stress tensor in the first-order Raviart-Thomas space. By combining the equilibrated residual method and the hypercircle method, it is shown that the error estimator gives a fully computable upper bound on the actual error. Numerical results are provided to confirm the theory and illustrate the effectiveness of our error estimator.The weak formulation for (1.1)-(1.2) seeks the displacement u ∈ H 1where (·, ·) Ω is the standard inner product in (L 2 (Ω)) d (d = 1, 2) and. We adopt the standard notation for the Sobolev space H k (S) over a set S equipped with the norm ∥ · ∥ k,S and semi-norm | · | k,S .Nowadays it is well established that one should apply adaptive mesh refinement based on a posteriori error estimators for efficient implementation of numerical methods. Various types of error estimators have been developed and successfully implemented for the linear elasticity problem; see, for example, the survey paper [1]. We are particularly interested in the error estimators which give fully computable upper bounds on the actual error without involving unknown constants. Such error estimators were constructed in [2] for the P 1 conforming element and in [3] for the P 2 conforming and nonconforming elements by combining the equilibrated residual method and the hypercircle method. The key step there was recovery of a symmetric H(div)-conforming approximation of the stress tensor σ in an appropriate finite element space of symmetric tensors from the equilibrated normal stress approximation, which is quite complicated due to the large dimension of the local finite element space used (the Arnold-Winther space in [2] and the Arnold-Douglas-Gupta space in [3]), although the computation is done locally on each element.In this paper we propose a new error estimator which requires much less computation than [2, 3], while achieving fully computable upper bounds on the actual error. This is accomplished by recovering a nonsymmetric H(div)-conforming approximation of the stress tensor σ in the nonsymmetric Raviart-Thomas space of first order whose local dimension is smaller and makes the implementation easier than the symmetric tensor spaces mentioned above. When compared with the ones from [2, 3], our estimator contains the additional contribution arising from the non-symmetry of the recovered stress tensor approximation and thus strongly depending on computable upper bounds on the constants of local Korn's inequality. A similar consideration was given to the Stokes problem in [4] but the derived estimator was not fully computable as the constant in the upper bound was not estimated.To fix ideas, we consider the stabilized P 1 nonconforming finite element proposed in [5] for which, unlike the P 1 conforming element, the equilibrated normal stress approximation is explicitly constructed without solving local linear sy...
