Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

TL;DR

This paper develops a numerical scheme combining Galerkin and semi-implicit Euler--Maruyama methods for stochastic evolution equations in Hilbert spaces, proving convergence in probability with explicit rates depending on regularity.

Contribution

It introduces a convergence proof for a combined space-time numerical scheme for stochastic evolution equations, including nonlinear heat and shell models, with explicit error rates.

Findings

Proves convergence in probability of the numerical scheme.

Provides explicit convergence rates depending on regularity parameter.

Applicable to models like GOY, Sabra shell models, and nonlinear heat equations.

Abstract

The present paper is devoted to the numerical approximation of an abstract stochastic nonlinear evolution equation in a separable Hilbert space {$\mathrm{H}$}. Examples of equations which fall into our framework include the GOY and Sabra shell models and { a class of nonlinear heat equations.} The space-time numerical scheme is defined in terms of a Galerkin approximation in space and a { semi-implicit Euler--Maruyama scheme in time}. {We prove the convergence in probability of our scheme by means of an estimate of the error on a localized set of arbitrary large probability.} Our error estimate is shown to hold in a more regular space $\mathrm{V}_{\beta}\subset \mathrm{H}$ with $\beta \in [0,\frac14)$ and { that the explicit rate of convergence of our scheme depends on this parameter $\beta$. }