*The gradient discretisation method (GDM) is a generic framework, covering many classical methods (Finite Elements, Finite Volumes, Discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for a general stochastic evolution problem based on a Leray--Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the Gradient Scheme (GS) solutions is proved by using Discrete Functional Analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way, we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.
翻译:* 梯度离散法(GDM)是一个通用框架,涵盖许多用于设计和分析扩散模型数字方法的古典方法(元素、有限量、不连续的Galerkin等)。在本文件中,我们研究GDM用于基于激光-Lion型操作员的一般随机进化问题。问题包括作为特定案例的Stochistic $p$-Laplace方程。渐进制(GS)解决方案的趋同通过使用分立功能分析技术、Skorohod 理论和 Kolmogorov 测试得到证明。特别是,我们独立证明这一问题存在薄弱的马丁醇溶液。这样,我们为证明相近的合成部分差异方程式的趋同打下基础并提供技术。