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

Stefan Metzger

TL;DR

This work develops and analyzes a convergent, linear, fully discrete finite element scheme for a stochastic diffuse-interface model of line tension, described by a stochastic Cahn–Hilliard equation with Allen–Cahn-type dynamic boundary conditions and multiplicative noise. An augmented SAV framework renders the scheme linear and energy-stable, and the authors prove existence of pathwise unique martingale solutions in the limit and convergence to strong solutions for a given Wiener process via a Gyöngy–Krylov framework. The analysis combines uniform a priori estimates, Nikolskii-type time regularity, Jakubowski’s Skorokhod representation, and a Gyöngy–Krylov argument to establish rigorous convergence and strong solvability. The results provide a solid mathematical foundation for reliable, pathwise numerical investigations of contact-line fluctuations in droplets with line-tension effects. The methodology offers a transferable approach for stochastic gradient-flow PDEs with dynamic boundaries and complex nonlinear energies.

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 Gyöngy--Krylov characterization of convergence in probability provides convergence towards strong solutions and thereby completes the proof.

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

TL;DR

This work develops and analyzes a convergent, linear, fully discrete finite element scheme for a stochastic diffuse-interface model of line tension, described by a stochastic Cahn–Hilliard equation with Allen–Cahn-type dynamic boundary conditions and multiplicative noise. An augmented SAV framework renders the scheme linear and energy-stable, and the authors prove existence of pathwise unique martingale solutions in the limit and convergence to strong solutions for a given Wiener process via a Gyöngy–Krylov framework. The analysis combines uniform a priori estimates, Nikolskii-type time regularity, Jakubowski’s Skorokhod representation, and a Gyöngy–Krylov argument to establish rigorous convergence and strong solvability. The results provide a solid mathematical foundation for reliable, pathwise numerical investigations of contact-line fluctuations in droplets with line-tension effects. The methodology offers a transferable approach for stochastic gradient-flow PDEs with dynamic boundaries and complex nonlinear energies.

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 Gyöngy--Krylov characterization of convergence in probability provides convergence towards strong solutions and thereby completes the proof.
Paper Structure (11 sections, 20 theorems, 203 equations)

This paper contains 11 sections, 20 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. The discrete scheme
  4. Main results
  5. Existence of discrete solutions
  6. Regularity results
  7. Compactness properties of discrete solutions
  8. Passage to the limit
  9. Pathwise uniqueness of martingale solutions
  10. Convergence towards strong solutions
  11. Conclusion

Key Result

Lemma 2.1

Let $\mathcal{T}_h$ and $\mathcal{T}_h^{\Gamma}$ satisfy item:space1 and item:space2, respectively. Furthermore, let $p\in[1,\infty)$, $1\leq q\leq \infty$ and $q^*=\tfrac{q}{q-1}$ for $q<\infty$ or $q^*=1$ for $q=\infty$. Then hold true for all $f_{h},g_{h}\in U_{h}^{\mathcal{O}}$ and $u_{h},v_{h}\in U_{h}^{\Gamma}$.

Theorems & Definitions (36)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • ...and 26 more