Additive Schwarz methods for fourth-order variational inequalities

TL;DR

This paper develops and analyzes scalable additive Schwarz methods for efficiently solving fourth-order variational inequalities, with theoretical proofs and numerical validation demonstrating their effectiveness.

Contribution

It introduces a unified framework for finite element methods and proves the scalability of a new two-level additive Schwarz method for these inequalities.

Findings

The two-level method's convergence depends only on $H/h$ and $H/ ext{overlap}$.

A novel nonlinear positivity-preserving coarse interpolation operator was constructed.

Numerical experiments confirm the theoretical scalability results.

Abstract

Fourth-order variational inequalities are encountered in various scientific and engineering disciplines, including elliptic optimal control problems and plate obstacle problems. In this paper, we consider additive Schwarz methods for solving fourth-order variational inequalities. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ measures the overlap among the subdomains. This proof relies on a new nonlinear positivity-preserving coarse interpolation operator, the construction of which was previously unknown. To the best of our knowledge, this analysis represents the first investigation into the scalability of the two-level additive Schwarz method for fourth-order variational inequalities. Our theoretical results are verified by numerical experiments.