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Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

Sebastian Becker, Benjamin Gess, Arnulf Jentzen, Peter E. Kloeden

TL;DR

This work establishes essentially sharp strong convergence rates for fully discrete numerical approximations of SPDEs driven by space-time white noise with superlinear nonlinearities, notably the stochastic Allen-Cahn equation. By employing a path-dependent Lyapunov-type function, the authors obtain uniform a priori moment bounds for the full-discrete scheme and perform a meticulous pathwise and $L^p$ error analysis using Galerkin projections and integrated nonlinearity. The main result provides explicit strong-convergence rates in time and space and proves corresponding lower bounds to confirm sharpness, while the stochastic Allen-Cahn specialization illustrates practical impact for a classic nonlinear SPDE. Additionally, the paper derives lower/upper bounds for linear stochastic heat equations to contextualize the bounds within the linear regime. Overall, the results offer a rigorous, general framework for strong convergence of full discretizations in SPDEs with superlinear growth, with concrete implications for numerical simulations of stochastic phase-field models.

Abstract

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities. In particular, in the scientific literature there exists neither a result which proves strong convergence rates nor a result which proves weak convergence rates for full-discrete numerical approximations of stochastic Allen-Cahn equations. In this article we bridge this gap and establish strong convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. Moreover, we also establish lower bounds for strong temporal and spatial approximation errors which demonstrate that our strong convergence rates are essentially sharp and can, in general, not be improved.

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

TL;DR

This work establishes essentially sharp strong convergence rates for fully discrete numerical approximations of SPDEs driven by space-time white noise with superlinear nonlinearities, notably the stochastic Allen-Cahn equation. By employing a path-dependent Lyapunov-type function, the authors obtain uniform a priori moment bounds for the full-discrete scheme and perform a meticulous pathwise and error analysis using Galerkin projections and integrated nonlinearity. The main result provides explicit strong-convergence rates in time and space and proves corresponding lower bounds to confirm sharpness, while the stochastic Allen-Cahn specialization illustrates practical impact for a classic nonlinear SPDE. Additionally, the paper derives lower/upper bounds for linear stochastic heat equations to contextualize the bounds within the linear regime. Overall, the results offer a rigorous, general framework for strong convergence of full discretizations in SPDEs with superlinear growth, with concrete implications for numerical simulations of stochastic phase-field models.

Abstract

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities. In particular, in the scientific literature there exists neither a result which proves strong convergence rates nor a result which proves weak convergence rates for full-discrete numerical approximations of stochastic Allen-Cahn equations. In this article we bridge this gap and establish strong convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. Moreover, we also establish lower bounds for strong temporal and spatial approximation errors which demonstrate that our strong convergence rates are essentially sharp and can, in general, not be improved.

Paper Structure

This paper contains 33 sections, 35 theorems, 375 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgments
  4. A priori bounds for the numerical approximation
  5. Pathwise error estimates
  6. Setting
  7. On the separability of a certain Banach space
  8. Analysis of the error between the Galerkin projection of the exact solution and the Galerkin projection of the semilinear integrated version of the numerical approximation
  9. Analysis of the error between the numerical approximation and the Galerkin projection of the semilinear integrated version of the numerical approximation
  10. Temporal regularity for the Galerkin projection of the semilinear integrated version of the numerical approximation
  11. Strong error estimates
  12. Setting
  13. Analysis of the error between the Galerkin projection of the exact solution and the Galerkin projection of the semilinear integrated version of the numerical approximation
  14. Analysis of the error between the numerical approximation and the Galerkin projection of the semilinear integrated version of the numerical approximation
  15. Temporal regularity for the Galerkin projection of the semilinear integrated version of the numerical approximation
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $T \in (0,\infty)$, $( H, \langle \cdot, \cdot \rangle_H, \left\| \cdot \right\|\!_H ) = ( L^2( (0,1); \mathbb{R} ), \langle \cdot, \cdot \rangle_{ L^2( (0,1); \mathbb{R} ) }, \left\| \cdot \right\|\!_{ L^2( (0,1) ; \mathbb{R} ) } )$, $a_0, a_1$, $a_2 \in \mathbb{R}$, $a_3 \in (-\infty,0]$, $(e_ and Then

Theorems & Definitions (68)

  • Theorem 1.1
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem:Y_a_priori']}
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:separable']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:Y_regularity']}
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['lem:pw_X_XX_diff']}
  • Lemma 3.4
  • ...and 58 more