A Strongly Convergent Primal-Dual Method for Nonoverlapping Domain Decomposition

TL;DR

This paper introduces a primal-dual parallel proximal splitting method for domain decomposition problems in PDEs, achieving strong convergence without restrictive assumptions, applicable to linear and nonlinear cases.

Contribution

It presents a novel, strongly convergent primal-dual method for nonoverlapping domain decomposition, capable of handling complex transmission conditions and broad problem classes.

Findings

Method converges strongly in energy spaces

Applicable to linear and nonlinear PDEs

Handles nonlinear interface conditions

Abstract

We propose a primal-dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling constraints which model various properties of the solution at the interfaces. The proposed method can handle a wide range of linear and nonlinear problems, with flexible, possibly nonlinear, transmission conditions across the interfaces. Strong convergence in the energy spaces is established in this general setting, and without any additional assumption on the energy functions or the geometry of the problem. Several examples are presented.