The gradient discretization 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.

中文翻译：

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Leray-Lions算子随机演化方程数值方法的设计与收敛性分析

梯度离散化方法 (GDM) 是一个通用框架，涵盖了许多经典方法（有限元、有限体积、不连续 Galerkin 等），用于设计和分析扩散模型的数值方案。在本文中，我们研究了基于 Leray-Lions 类型算子的一般随机演化问题的 GDM。该问题包含作为特例的随机 $p$-Laplace 方程。通过使用离散泛函分析技术、Skorohod 定理和 Kolmogorov 检验证明了梯度方案 (GS) 解的收敛性。特别是，我们提供了该问题存在弱鞅解的独立证明。通过这种方式，我们为证明 GS 近似随机偏微分方程的收敛性奠定了基础并提供了技术。