On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation

TL;DR

This paper establishes an explicit convergence rate of .5.5 for Yosida approximations of the nonlocal Cahn-Hilliard equation, enhancing understanding of solution approximation and error estimation.

Contribution

It provides the first explicit convergence rate for Yosida approximations in the nonlocal Cahn-Hilliard equation using maximal monotone operator theory.

Findings

Convergence rate of .5.5.5 for Yosida approximation.

Nonlocal operator is of Hilbert-Schmidt type.

Results applicable to Galerkin method error estimates.

Abstract

It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda}$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $\lambda$ could be linked to the discretization parameters, yielding appropriate error estimates.