A convergent augmented SAV scheme for stochastic Cahn--Hilliard equations with dynamic boundary conditions describing contact line tension

TL;DR

This paper introduces a convergent numerical scheme for stochastic Cahn--Hilliard equations with dynamic boundary conditions, incorporating thermal fluctuations and proving convergence to strong solutions.

Contribution

It develops a new augmented scalar auxiliary variable scheme that ensures convergence and well-posedness for stochastic Cahn--Hilliard equations with boundary contact line tension.

Findings

Proved existence of pathwise unique strong solutions.

Established convergence of the numerical scheme to these solutions.

Provided energy estimates and regularity results for the scheme.

Abstract

We augment a thermodynamically consistent diffuse interface model for the description of line tension phenomena by multiplicative stochastic noise to capture the effects of thermal fluctuations and establish the existence of pathwise unique (stochastically) strong solutions. By starting from a fully discrete linear finite element scheme, we do not only prove the well-posedness of the model, but also provide a practicable and convergent scheme for its numerical treatment. Conceptually, our discrete scheme relies on a recently developed augmentation of the scalar auxiliary variable approach, which reduces the requirements on the time regularity of the solution. By showing that fully discrete solutions to this scheme satisfy an energy estimate, we obtain first uniform regularity results. Establishing Nikolskii estimates with respect to time, we are able to show convergence towards pathwise unique martingale solutions by applying Jakubowski's generalization of Skorokhod's theorem. Finally, a generalization of the Gy\"ongy--Krylov characterization of convergence in probability provides convergence towards strong solutions and thereby completes the proof.