Errors of finite element method for perfectly elasto-plastic problems

This paper discusses convergence of the finite element method for variational problems of the Hencky plasticity theory. To obtain a priori convergence estimates we use the method of ''double approximation''. In the framework of this approach perfectly elastoplastic problem is approximated by some regularized problem. Hence, finite element solutions of the regularized problem depend on the regularization parameter delta and the mesh parameter h. For these solutions we obtain a projection type error estimate. This estimate is a sum of the two parts which represent the errors of regularization and discretization, respectively. Then we prove that under some assumptions on the external data the minimizer of the regularized problem and the maximizer of its dual problem possess additional differentiability properties and deduce the corresponding estimates which explicitly depend on the parameter delta. This makes it possible to prove that there is a dependence between delta and h such that piecewise-affine continuous approximations of the regularized problems generate a sequence of tensor valued functions which converges to the exact solution of the Hencky plasticity problem.