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Convergence of a double step scheme for a class of second order Clarke subdifferential inclusions

Krzysztof Bartosz, Paweł Szafraniec

TL;DR

The paper addresses second-order evolution inclusions with a Clarke subdifferential term and develops a double-step Rothe time-discretization scheme. It proves solvability of the Rothe problem, establishes uniform a priori bounds, and demonstrates weak convergence of semidiscrete solutions to a weak solution of the original problem, thereby extending convergence theory for nonsmooth dynamic models. The approach is applicable to dynamic viscoelastic contact problems with nonsmooth friction, connecting hemivariational inequalities with subdifferential inclusions. This work provides a rigorous numerical-analytic foundation for using double-step Rothe schemes in second-order nonsmooth evolution problems. Overall, it advances the mathematical justification for discretization-based analysis of Clarke subdifferential inclusions in mechanics.

Abstract

In this paper we deal with a second order evolution inclusion involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a double step time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem.

Convergence of a double step scheme for a class of second order Clarke subdifferential inclusions

TL;DR

The paper addresses second-order evolution inclusions with a Clarke subdifferential term and develops a double-step Rothe time-discretization scheme. It proves solvability of the Rothe problem, establishes uniform a priori bounds, and demonstrates weak convergence of semidiscrete solutions to a weak solution of the original problem, thereby extending convergence theory for nonsmooth dynamic models. The approach is applicable to dynamic viscoelastic contact problems with nonsmooth friction, connecting hemivariational inequalities with subdifferential inclusions. This work provides a rigorous numerical-analytic foundation for using double-step Rothe schemes in second-order nonsmooth evolution problems. Overall, it advances the mathematical justification for discretization-based analysis of Clarke subdifferential inclusions in mechanics.

Abstract

In this paper we deal with a second order evolution inclusion involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a double step time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem.
Paper Structure (5 sections, 19 theorems, 143 equations)

This paper contains 5 sections, 19 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Problem formulation
  4. The Rothe problem
  5. Convergence

Key Result

Proposition 2.4

Let $X$ and $U$ be two reflexive Banach spaces and $\iota\colon X\to U$ a linear, continuous and compact operator. Let $J\colon U\to\mathbb{R}$ be a locally Lipschitz functional and assume that its Clarke subdifferential satisfies with $c>0$. Then the multivalued operator $M\colon X\to 2^{X^*}$ defined by is pseudomonotone.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 13 more