Application of functional error estimates with mixed approximations to plane problems of linear elasticity

S. I. Repin and his colleagues' studies addressing functional a posteriori error estimates for solutions of linear elasticity problems are further developed. Although the numerical results obtained for planar problems by A. V. Muzalevsky and Repin point to advantages of the adaptive approach used, the degree of overestimation of the absolute error increases noticeably with mesh refinement. This shortcoming is eliminated by using approximations typical of mixed finite element methods. A comparative analysis is conducted for the classical finite element approximations, mixed Raviart–Thomas approximations, and relatively recently proposed Arnold–Boffi–Falk mixed approximations. It is shown that the last approximations are the most efficient.