Monika Weymuth | Searcharxiv

arXiv2020

A Priori Error Analysis for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

We consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. Based on a quadratic growth condition we derive nearly optimal a priori error estimates. Moreover, we establish second order sufficient optimality conditions that ensure a quadratic growth condition. These conditions are rather restrictive, but allow us to construct a one-dimensional locally optimal solution with reduced regularity, which serves as an exact solution for numerical experiments.

math.NA↗

arXiv2017

Adaptive Local (AL) Basis for Elliptic Problems with $L^\infty$-Coefficients

We define a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. An adaptive local finite element basis (AL basis) on a coarse mesh which does not resolve the matrix of the media is constructed by solving finite-dimensional localized problems. The method requires $O(log(1/H)^{d+1})$ basis functions per mesh point. We prove that the optimal finite element convergence rates are preserved.

math.NA↗