Convergence Analysis of the Nonoverlapping Robin-Robin Method for Nonlinear Elliptic Equations

TL;DR

This paper proves the convergence of the Robin-Robin domain decomposition method for nonlinear elliptic equations with p-structure, extending previous linear results to nonlinear cases using new operator theory.

Contribution

It develops a new theoretical framework for nonlinear Steklov-Poincaré operators based on p-structure, enabling convergence analysis of the Robin-Robin method for nonlinear elliptic equations.

Findings

Proves convergence for nonlinear elliptic equations with p-structure.

Extends domain decomposition convergence theory to nonlinear cases.

Framework applicable on Lipschitz domains without regularity restrictions.

Abstract

We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a $p$-structure, including degenerate diffusion equations governed by the $p$-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincar\'e operators based on the $p$-structure and the $L^p$-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.