# An Integral Equation Method for the Cahn-Hilliard Equation in the Wetting Problem

## TL;DR

This paper introduces an efficient integral equation method for solving the Cahn-Hilliard equation with boundary conditions modeling wetting, utilizing advanced numerical techniques for high accuracy and linear complexity.

## Contribution

It develops a novel integral equation approach combining volume potentials, SKIE, and QBX with FMM acceleration for the Cahn-Hilliard equation in wetting problems.

## Key findings

- Achieves high-order convergence with adaptive refinement.
- Linear complexity in degrees of freedom.
- Effective handling of boundary conditions with prescribed Young's angles.

## Abstract

We present an integral equation approach to solving the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. The discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, we split the solution into a volume potential computed with free space kernels, plus the solution to a second kind integral equation (SKIE). The volume potential is evaluated with the help of a box-based volume-FMM method. For non-box domains, source density is extended by solving a biharmonic Dirichlet problem. The near-singular boundary integrals are computed using quadrature by expansion (QBX) with FMM acceleration. Our method has linear complexity in the number of surface/volume degrees of freedom and can achieve high order convergence with adaptive refinement to manage error from function extension.