A mixed finite element method for the stochastic Boussinesq equations with multiplicative noise

TL;DR

This paper develops and analyzes a fully discrete mixed finite element method for the stochastic Boussinesq equations with multiplicative noise, providing convergence proofs and numerical validation.

Contribution

It introduces a novel combination of mixed finite element spatial discretization with semi-implicit Euler-Maruyama temporal scheme for stochastic Boussinesq equations, with proven error bounds and convergence.

Findings

Error bounds for velocity, pressure, and temperature approximations

Convergence in probability in $L^2$ and $H^1$ norms

Numerical experiments confirming theoretical results

Abstract

This work investigates a fully discrete mixed finite element method for the stochastic Boussinesq system driven by multiplicative noise. The spatial discretization is performed using a standard mixed finite element method, while the temporal discretization is based on a semi-implicit Euler-Maruyama scheme. By combining a localization technique with high-moment stability estimates, we establish error bounds for the velocity, pressure, and temperature approximations. As a direct consequence, we prove convergence in probability for the fully discrete method in both $L^2$ and $H^1$-type norms. Several numerical experiments are presented to validate the theoretical error estimates and demonstrate the effectiveness of the proposed scheme.