We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of It\^o. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gy\"{o}ngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.

翻译：我们在这里研究由Lipschitz 持续多复制性噪音所强制的热方程的有限量计划近似值。 更确切地说, 我们考虑的是时间上半隐含的离散和空间上两点通量近似计划。 我们根据Prokhorov 的理论对方法进行了调整, 以取得分布结果的趋同, 然后Skorokhod 的代言词使这个计划趋于趋同于马丁格尔解决方案, 而Gy\"{o}ngy- Krylov 的论则被用来证明这个计划与我们抛物法问题独特的不同解决方案的趋同性。