Green’s function representation and numerical approximation of the two-dimensional stochastic Stokes equation

Abstract

This paper investigates the two-dimensional unsteady Stokes equation with general additive noise. The primary contribution is the derivation of the relevant estimate of Green’s tensor, which provides a fundamental representation for the solution of this stochastic equation. We demonstrate the crucial role of Green’s function in understanding the stability and perturbation characteristics of the stochastic Stokes system. Furthermore, we analyze the convergence properties of the Euler–Maruyama (EM) scheme for temporal discretization and derive error estimates for a Galerkin finite element discretization using the Taylor–Hood method for spatial approximation. This work provides a strong convergence of order $O\left(h{\left(\Delta t\right)}^{-\frac{1}{2}}+\Delta t\right)$ of the velocity in the $L^2(0,T;L^2\left(D\right))$ norm and $O\left(h{\left(\Delta t\right)}^{-\frac{3}{2}}+\Delta t\right)$ of the pressure in the $L^2(0,T;H^{-1}\left(D\right))$ norm based on the Green tensor approach. These results contribute to a deeper understanding of the stochastic behavior of fluid dynamics systems, paving the way for improved theoretical modeling and more accurate numerical simulations in diverse fields such as meteorology, oceanography, and engineering applications.