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Strong approximation of the time-fractional Cahn--Hilliard equation driven by a fractionally integrated additive noise

Mariam Al-Maskari, Samir Karaa

TL;DR

This work analyzes strong convergence of numerical schemes for a time-fractional stochastic Cahn–Hilliard equation driven by fractionally integrated additive noise. It develops a spatial discretization via piecewise linear finite elements and a temporal discretization by convolution quadrature for both ${^C}\partial_t^{\alpha}$ and the fractional integral, with an $L^2$-projection of the noise, and provides rigorous error estimates via energy methods and Hölder regularity of the solution. The authors prove convergence rates for both semidiscrete and fully discrete schemes that depend on the spatial mesh size $h$, time step $\tau$, and noise regularity (encoded by $\beta$, $\alpha$, $\gamma$), and they address the analytical challenges posed by the unbounded elliptic operator. Numerical experiments in one dimension validate the theoretical rates and illustrate the influence of noise regularity on convergence behavior.

Abstract

In this paper, we consider the numerical approximation of a time-fractional stochastic Cahn--Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order $α\in(0,1)$ and a fractional time-integral noise of order $γ\in[0,1]$. The numerical scheme approximates the model by a piecewise linear finite element method in space and a convolution quadrature in time (for both time-fractional operators), along with the $L^2$-projection for the noise. We carefully investigate the spatially semidiscrete and fully discrete schemes, and obtain strong convergence rates by using clever energy arguments. The temporal Hölder continuity property of the solution played a key role in the error analysis. Unlike the stochastic Allen--Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model adds complexity and challenges to the error analysis. To overcome these difficulties, several new techniques and error estimates are developed. The study concludes with numerical examples that validate the theoretical findings.

Strong approximation of the time-fractional Cahn--Hilliard equation driven by a fractionally integrated additive noise

TL;DR

This work analyzes strong convergence of numerical schemes for a time-fractional stochastic Cahn–Hilliard equation driven by fractionally integrated additive noise. It develops a spatial discretization via piecewise linear finite elements and a temporal discretization by convolution quadrature for both and the fractional integral, with an -projection of the noise, and provides rigorous error estimates via energy methods and Hölder regularity of the solution. The authors prove convergence rates for both semidiscrete and fully discrete schemes that depend on the spatial mesh size , time step , and noise regularity (encoded by , , ), and they address the analytical challenges posed by the unbounded elliptic operator. Numerical experiments in one dimension validate the theoretical rates and illustrate the influence of noise regularity on convergence behavior.

Abstract

In this paper, we consider the numerical approximation of a time-fractional stochastic Cahn--Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order and a fractional time-integral noise of order . The numerical scheme approximates the model by a piecewise linear finite element method in space and a convolution quadrature in time (for both time-fractional operators), along with the -projection for the noise. We carefully investigate the spatially semidiscrete and fully discrete schemes, and obtain strong convergence rates by using clever energy arguments. The temporal Hölder continuity property of the solution played a key role in the error analysis. Unlike the stochastic Allen--Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model adds complexity and challenges to the error analysis. To overcome these difficulties, several new techniques and error estimates are developed. The study concludes with numerical examples that validate the theoretical findings.
Paper Structure (12 sections, 21 theorems, 128 equations, 4 tables)

This paper contains 12 sections, 21 theorems, 128 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Norms and operators
  4. The Wiener process
  5. Representation of the Solution
  6. Stochastic convolution
  7. Regularity of the Solution
  8. Galerkin Semidiscrete Problem
  9. Fully Discrete Scheme
  10. Numerical Results
  11. Implementation of the noise term
  12. Numerical tests and discussions

Key Result

Lemma 2.1

Let $u_0\in \dot{H}^\nu(\Omega)$. Then, for $m\geq 1$ and $0\leq q,\nu\leq 4$, where for $\ell=0,\, 0\leq \nu\leq q\leq 4$ .

Theorems & Definitions (39)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 4.1
  • Lemma 4.2
  • ...and 29 more