We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) that include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak formulation and provide numerical simulations to demonstrate the practicability of the proposed discretization.

中文翻译：

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奇异简并抛物型随机偏微分方程的数值逼近

我们研究一类通用的奇异简并抛物型随机偏微分方程（SPDE），其中特别包括随机多孔介质方程和随机快速扩散方程。我们基于非常弱的公式提出了所考虑的 SPDE 的完全离散数值近似。通过利用所提出的公式的单调性特性，我们证明了数值近似对唯一解的收敛性。此外，我们构建了一个可实现的有限元方案，用于非常弱的公式的空间离散化，并提供数值模拟来证明所提出的离散化的实用性。