Numerical analysis of the primal problem of elastoplasticity with hardening

The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evoluti on problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequaliti es corresponding to the primal formulation of (Hencky) plasticity. The fini te element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear reap. piecewise constant trial functions. An a priori error estimate is established for the fully-discret e method which shows linens convergence as the mesh-size tends to zero, pro vided the exact displacement field u is smooth. Near the boundary of the pl astic domain, which is unknown a priori, it is most likely that 21 is nonsm ooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error esti mate. This estimate is reliable and efficient in the sense that the quotien t of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening.