Numerical analysis of primal elastoplasticity with hardening

The quasi-static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered in its primal formulation with emphasis on improved optimal convergence of the lowest order scheme. Within one time-step of an implicit time-discretisation, the finite element method leads to a minimisation problem for non-smooth convex functions on discrete subspaces. The internal variables can be eliminated such that the displacement is the only remaining variable. An priori error estimate is presented for the fully-discrete method which proves linear convergence in space. Further, an a posteriori error estimate justifies an automatic adaptive mesh-refining algorithm. Numerical experiments confirm our theoretical predictions and the superiority of the adapted mesh.