Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem

TL;DR

This paper develops and analyzes an iterative Schwarz coupling method for efficiently solving a 2-D Laplace problem coupled with a 1-D model, providing convergence proofs and numerical validation.

Contribution

It introduces a novel Schwarz-like iterative coupling approach for dimensionally heterogeneous problems, with theoretical convergence analysis and interface placement insights.

Findings

Proves convergence of the coupling algorithm.

Provides guidelines for interface placement.

Numerically validates theoretical results.

Abstract

In the present work, we study and analyze an efficient iterative coupling method for a dimensionally heterogeneous problem . We consider the case of 2-D Laplace equation with non symmetric boundary conditions with a corresponding 1-D Laplace equation. We will first show how to obtain the 1-D model from the 2-D one by integration along one direction, by analogy with the link between shallow water equations and the Navier-Stokes system. Then, we will focus on the design of an Schwarz-like iterative coupling method. We will discuss the choice of boundary conditions at coupling interfaces. We will prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and the control of the difference between a global 2-D reference solution and the 2-D coupled one. These theoretical results will be illustrated numerically.