Analysis of Chorin-Type Projection Methods for the Stochastic Stokes Equations with General Multiplicative Noises

TL;DR

This paper analyzes two Chorin-type projection methods for stochastic Stokes equations with multiplicative noise, establishing optimal convergence rates for velocity and pressure, and examining the impact of spatial discretization errors through numerical validation.

Contribution

It introduces a modified Chorin scheme employing Helmholtz decomposition for stochastic noise, providing optimal error estimates and analyzing the deterioration of the standard scheme as time step decreases.

Findings

Optimal convergence rates for velocity and pressure approximations.

Suboptimal spatial error estimates with growth factor in error constants.

Numerical validation confirming theoretical error bounds.

Abstract

This paper is concerned with numerical analysis of two fully discrete Chorin-type projection methods for the stochastic Stokes equations with general non-solenoidal multiplicative noise. The first scheme is the standard Chorin scheme and the second one is a modified Chorin scheme which is designed by employing the Helmholtz decomposition on the noise function at each time step to produce a projected divergence-free noise and a "pseudo pressure" after combining the original pressure and the curl-free part of the decomposition. Optimal order rates of the convergence are proved for both velocity and pressure approximations of these two (semi-discrete) Chorin schemes. It is crucial to measure the errors in appropriate norms. The fully discrete finite element methods are formulated by discretizing both semi-discrete Chorin schemes in space by the standard finite element method. Suboptimal order error estimates are derived for both fully discrete methods. It is proved that all spatial error constants contain a growth factor $k^{-1/2}$, where $k$ denotes the time step size, which explains the deteriorating performance of the standard Chorin scheme when $k\to 0$ and the space mesh size is fixed as observed earlier in the numerical tests of [9]. Numerical results are also provided to guage the performance of the proposed numerical methods and to validate the sharpness of the theoretical error estimates.