Convergence of substructuring Methods for the Cahn-Hilliard Equation

TL;DR

This paper develops and analyzes substructuring algorithms, specifically Dirichlet-Neumann and Neumann-Neumann methods, for solving the nonlinear Cahn-Hilliard equation across multiple domains, demonstrating their convergence through numerical experiments.

Contribution

It introduces and studies the convergence of domain decomposition methods for the Cahn-Hilliard equation, extending to multi-subdomain settings.

Findings

Convergence of DN and NN methods for the Cahn-Hilliard equation is established.

Numerical results confirm the theoretical convergence behavior.

Methods are applicable in 1D and 2D domain decompositions.

Abstract

In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and since then it has appeared in many fields ranging from tumour growth simulation, image processing, thin liquid films, population dynamics etc. Being a non-linear equation, it is important to develop robust numerical techniques to solve the CH equation. Here we present the formulation of Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to CH equation and study their convergence behaviour. We consider the domain-decomposition based DN and NN methods in one and two space dimension for two subdomains and extend the study for multi-subdomain setting for NN method. We verify our findings with numerical results.