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Convergence of double step scheme for a class of parabolic Clarke subdifferential inclusions

Krzysztof Bartosz, Paweł Szafraniec, Jing Zhao

TL;DR

This work analyzes a parabolic evolution inclusion with a multivalued Clarke subdifferential term $\partial J$, and develops a double-step Rothe time discretization to approximate its solutions. It establishes solvability and a priori bounds for the Rothe scheme and proves that the semidiscrete solutions converge weakly to a weak solution of the original problem, using pseudomonotone operator theory, compactness (Aubin–Celina type), and upper semicontinuity of the Clarke subdifferential. The main contributions are a rigorous convergence result for a higher-order time discretization of parabolic hemivariational inequalities and a concrete PDE example illustrating applicability to boundary nonlinearities defined by $\partial J$. The results provide a theoretically sound numerical approach for nonmonotone evolution problems in mechanics and related fields.

Abstract

In this paper we deal with a first 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 double step scheme for a class of parabolic Clarke subdifferential inclusions

TL;DR

This work analyzes a parabolic evolution inclusion with a multivalued Clarke subdifferential term , and develops a double-step Rothe time discretization to approximate its solutions. It establishes solvability and a priori bounds for the Rothe scheme and proves that the semidiscrete solutions converge weakly to a weak solution of the original problem, using pseudomonotone operator theory, compactness (Aubin–Celina type), and upper semicontinuity of the Clarke subdifferential. The main contributions are a rigorous convergence result for a higher-order time discretization of parabolic hemivariational inequalities and a concrete PDE example illustrating applicability to boundary nonlinearities defined by . The results provide a theoretically sound numerical approach for nonmonotone evolution problems in mechanics and related fields.

Abstract

In this paper we deal with a first 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 (6 sections, 15 theorems, 118 equations)

This paper contains 6 sections, 15 theorems, 118 equations.

Table of Contents

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

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 (19)

  • 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 9 more