Convergence of Rothe scheme for hemivariational inequalities of parabolic type

TL;DR

This paper analyzes the convergence of the Rothe method for hemivariational inequalities of parabolic type, providing new theoretical insights and numerical validation for solving complex evolution PDEs with multivalued terms.

Contribution

It offers a unified convergence analysis framework for Rothe schemes applied to hemivariational inequalities, extending existing results to more general Banach space settings.

Findings

Proves convergence of the Rothe method for the problem class

Provides improved convergence results under certain conditions

Includes numerical examples demonstrating the method's effectiveness

Abstract

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^*\partial J(\iota u(t))\ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition the results on improved convergence as well as the numerical examples are presented.