Guaranteed a posteriori error estimator for mixed finite element methods of linear elasticity with weak stress symmetry

摘要

In this paper we propose an a posteriori error estimator for the mixed finite element methods of the linear elasticity problem with the symmetry condition weakly imposed on the stress tensor. The error estimator is constructed by making a proper decomposition of the stress error and using an argument similar to the hypercircle method. It is shown that the resulting estimator yields a guaranteed upper bound on the stress error which relies on computable upper bounds of the constants in the first and second Korn inequalities. We also establish the local lower bound by using the discrete Friedrichs inequality. Our approach is equivalent to the Helmholtz decomposition of the stress error but requires assumptions neither on the regularity of the solution nor the geometry of the domain. Numerical results are provided to illustrate the efficiency of our error estimator.