$hp$-FEM for Elastoplasticity & $hp$-Adaptivity Based on Local Error Reductions

Abstract: 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.