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Numerical approximation of the stochastic Cahn-Hilliard equation with space-time white noise near the sharp interface limit

Ľubomír Baňas, Jean Daniel Mukam

Abstract

We consider the stochastic Cahn-Hilliard equation with additive space-time white noise $ε^γ\dot{W}$ in dimension $d=2,3$, where $ε>0$ is an interfacial width parameter. We study numerical approximation of the equation which combines a structure preserving implicit time-discretization scheme with a discrete approximation of the space-time white noise. We derive a strong error estimate for the considered numerical approximation which is robust with respect to the inverse of the interfacial width parameter $ε$. Furthermore, by a splitting approach, we show that for sufficiently large scaling parameter $γ$, the numerical approximation of the stochastic Cahn-Hilliard equation converges uniformly to the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp interface limit $ε\rightarrow 0$.

Numerical approximation of the stochastic Cahn-Hilliard equation with space-time white noise near the sharp interface limit

Abstract

We consider the stochastic Cahn-Hilliard equation with additive space-time white noise in dimension , where is an interfacial width parameter. We study numerical approximation of the equation which combines a structure preserving implicit time-discretization scheme with a discrete approximation of the space-time white noise. We derive a strong error estimate for the considered numerical approximation which is robust with respect to the inverse of the interfacial width parameter . Furthermore, by a splitting approach, we show that for sufficiently large scaling parameter , the numerical approximation of the stochastic Cahn-Hilliard equation converges uniformly to the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp interface limit .
Paper Structure (11 sections, 28 theorems, 269 equations)

This paper contains 11 sections, 28 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Function spaces
  4. Existence and regularity results
  5. Numerical approximation
  6. Error analysis
  7. Sharp-interface limit
  8. Well-posedness of the numerical schemes \ref{['randompdescheme']} and \ref{['linearscheme']}
  9. $\mathbb{L}^\infty$-estimates for the solution of \ref{['linearscheme']} and the solution of \ref{['scheme1b']}
  10. $\mathbb{L}^{\infty}$-error estimate
  11. Convergence to the sharp-interface limit

Key Result

Proposition 2.1

(Debussche1& BM24) Let the initial value $u^{\varepsilon}_0$ be $\mathcal{F}_0$-measurable and $u^{\varepsilon}_0\in \mathbb{H}^{-1}$, then model1 has a unique strong variational solution, i.e., there exists a unique stochastic process $u\in C([0, T], \mathbb{H}^{-1})$$\mathbb{P}$-a.s., such that fo In addition, the solution $u\in L^2\left(\Omega, \{\mathcal{F}\}_t,\mathbb{P}; C([0, T]; \mathbb{H}

Theorems & Definitions (56)

  • Proposition 2.1
  • Lemma 2.1
  • Remark 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.2
  • Lemma 3.4
  • ...and 46 more