Numerical Analysis of Time-Depending PrimalElastoplasticity with Hardening

The quasi-static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme. In each time-step of a generalized midpoint scheme such as the implicit Euler or the Crank--Nicolson scheme, the spatial discretization consists of minimizing a convex but nonsmooth function on a subspace of continuous piecewise linear, resp., piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which, for smooth data and a smooth exact solution, proves linear convergence as the mesh-size tends to zero. Strong convergence of the time-derivatives is established under mild conditions on the mesh- and time-step sizes. Numerical experiments confirm our theoretical predictions on the improved spatial convergence and indicate that the Crank--Nicolson scheme is not always superior over the implicit Euler scheme in practice.