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

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 $\beta \in [0,\frac{1}{4})$ and { that the explicit rate of convergence of our scheme depends on this parameter $\beta$. }

References in corpus (3)

• Shell Models of Magnetohydrodynamic Turbulence
• Rate of Convergence of Space Time Approximations for stochastic evolution equations
• Splitting up method for the 2D stochastic Navier-Stokes equations

Cited by in corpus (5)

• On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Temporal approximation of stochastic evolution equations with irregular nonlinearities
• Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
• A splitting semi-implicit method for stochastic incompressible Euler equations on ${\mathbb{T}}^2$