  The first part of the cumulative thesis contains the numerical analysis of
different $hp$-finite element discretizations related to two different weak
formulations of a model problem in elastoplasticity with linearly kinematic
hardening. Thereby, the weak formulation either takes the form of a variational
inequality of the second kind, including a non-differentiable plasticity
functional, or represents a mixed formulation, in which the non-smooth
plasticity functional is resolved by a Lagrange multiplier. As the
non-differentiability of the plasticity functional causes many difficulties in
the numerical analysis and the computation of a discrete solution it seems
advantageous to consider discretizations of the mixed formulation. In a first
work, an a priori error analysis of an higher-order finite element
discretization of the mixed formulation (explicitly including the
discretization of the Lagrange multiplier) is presented. The relations between
the three different $hp$-discretizations are studied in a second work where
also a reliable a posteriori error estimator that also satisfies some (local)
efficiency estimates is derived. In a third work, an efficient semi-smooth
Newton solver is proposed, which is obtained by reformulating a discretization
of the mixed formulation as a system of decoupled nonlinear equations. The
second part of the thesis introduces a new $hp$-adaptive algorithm for solving
variational equations, in which the automatic mesh refinement does not rely on
the use of an a posteriori error estimator or smoothness indicators but is
based on comparing locally predicted error reductions.
