We address an original approach for the convergence analysis of a finite-volume scheme for the approximation of a stochastic diffusion-convection equation with multiplicative noise in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$) and with homogeneous Neumann boundary conditions. The idea behind our approach is to avoid using the stochastic compactness method. We study a numerical scheme that is semi-implicit in time and in which the convection and the diffusion terms are respectively approximated by means of an upwind scheme and the so called two-point flux approximation scheme (TPFA). By adapting well-known methods for the time discretization of stochastic PDEs and combining them with deterministic techniques applied to spatial discretization, we show strong convergence of our scheme towards the unique variational solution of the continuous problem in $L^p(0,T;L^2(\Omega;L^2(\Lambda)))$, for any finite $p\geq 1$.

翻译：本文提出了一种新颖的方法来分析有限体积格式的收敛性，该格式用于在有界域 $\mathbb{R}^d$（$d=2$ 或 $3$）中具有齐次 Neumann 边界条件的乘性噪声随机扩散-对流方程的近似。我们方法的核心思想是避免使用随机紧致性方法。我们研究了一种时间半隐式的数值格式，其中对流项和扩散项分别通过迎风格式和所谓的两点通量近似格式（TPFA）进行近似。通过将求解随机偏微分方程时间离散化的经典方法与应用于空间离散化的确定性技术相结合，我们证明了该格式对于任意有限 $p\geq 1$，在 $L^p(0,T;L^2(\Omega;L^2(\Lambda)))$ 意义下强收敛于连续问题的唯一变分解。