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Miriam Schönauer

Publications and source records attributed to Miriam Schönauer.

6 recordsLinked to original sources
arXiv2026

A posteriori error estimates for variational inequalities of the second kind in an abstract framework

In this paper, an abstract framework is introduced for the derivation of reliable a posteriori error estimators for variational inequalities of the second kind. Under certain conditions (local) efficiency of the error estimators can also be derived. The framework is based on the estimation of a residuum which results from an equivalent mixed formulation and from the replacement of the Lagrange multiplier by its (possibly post-processed) discrete counterpart. The approach is applied to an idealized frictional contact problem and a Mosolov type model problem in an $hp$-finite element discretization setup. Several numerical experiments demonstrate the broad applicability of the theoretical findings.

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arXiv2026

hp-Finite Elements for Elastoplasticity

This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational formulation is introduced to resolve the non-differentiablility of the so-called plasticity functional appearing in the weak formulation of the model problem as a variational inequality of the second kind. The discretization of the mixed formulation is then represented as a system of decoupled nonlinear equations which allows the application of an efficient semismooth Newton solver. Finally, an a priori and a posteriori error analysis is given.

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arXiv2025

A Semismooth Newton Solver and its application to an $hp$-FE Discretization in Elastoplasticity

In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and prove its well-definedness and local convergence. Thereby, the analysis heavily relies on a special eigenvalue interplay of two matrices involved in the considered nonlinear system. Next, we apply the general results to an $hp$-finite element discretization of a problem in elastoplasticity, which can be formulated as a system of nonlinear equations of the above type by using biorthogonal basis functions. Finally, numerical examples demonstrate the applicability and robustness of the proposed semismooth Newton solver with respect to $h$ and $p$.

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arXiv2024

Error estimates for perturbed variational inequalities of the first kind

In this paper, we derive a priori error estimates for variational inequalities of the first kind in an abstract framework. This is done by combining the first Strang Lemma and the Falk Theorem. The main application consists in the derivation of a priori error estimates for Galerkin methods, in which "variational crimes" may perturb the underlying variational inequality. Different types of perturbations are incorporated into the abstract framework and discussed by various examples. For instance, the perturbation caused by an inexact quadrature is examined in detail for the Laplacian obstacle problem. For this problem, guaranteed rates for the approximation error resulting from the use of higher-order finite elements are derived. In numerical experiments, the influence of the number of quadrature points on the approximation error and on the quadrature-related error itself is studied for several discretization methods.

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arXiv2024

On an optimal AFEM for elastoplasticity

In this paper, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [Comput. Math. Appl., 67(6) (2014), 1195-1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [Numer. Math., 132(1) (2016), 131-154], which presents an alternative approach to optimality without explicitly relying on the axioms.

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