Numerical analysis of time-depending primal elastoplasticity with hardening

The quasi-static elastoplastic evolution problem with combined isotropic an d kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme. In each time-step of a generalized midpoint schem e such as the implicit Euler or the Crank-Nicolson scheme, the spatial disc retization consists of minimizing a convex but nonsmooth function on a subs pace of continuous piecewise linear, resp., piecewise constant trial functi ons. An a priori error estimate is established for the fully-discrete metho d which, for smooth data and a smooth exact solution, proves linear converg ence as the mesh-size tends to zero. Strong convergence of the time-derivat ives is established under mild conditions on the mesh- and time-step sizes. Numerical experiments con rm our theoretical predictions on the improved s patial convergence and indicate that the Crank-Nicolson scheme is not alway s superior over the implicit Euler scheme in practice.