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.

翻译：本综合论文的第一部分包含与线运动硬化弹塑性模型问题两种不同弱形式相关的不同$hp$-有限元离散的数值分析。其中，弱形式要么采取第二类变分不等式的形式，包含不可微的塑性泛函，要么代表一种混合形式，其中非光滑塑性泛函通过拉格朗日乘子求解。由于塑性泛函的不可微性在数值分析和离散解计算中造成诸多困难，考虑混合形式的离散化似乎更为有利。在第一项工作中，提出了混合形式（明确包含拉格朗日乘子的离散化）的高阶有限元离散的先验误差分析。第二项工作研究了三种不同$hp$-离散之间的关系，并推导了一个可靠的后验误差估计器，该估计器也满足某些（局部）效率估计。第三项工作提出了一种高效的半光滑牛顿求解器，该求解器通过将混合形式的离散重新表述为解耦的非线性方程组系统而获得。论文第二部分引入了一种新的$hp$-自适应算法来解决变分方程，其中自动网格细化不依赖于后验误差估计器或光滑性指标的使用，而是基于比较局部预测的误差降低。