Numerical approximation of singular-degenerate parabolic stochastic PDEs

TL;DR

This paper develops a fully discrete numerical scheme for singular degenerate parabolic stochastic PDEs, including porous medium and fast diffusion equations, proving convergence and demonstrating practical implementation through numerical simulations.

Contribution

It introduces a novel fully discrete finite element scheme based on a very weak formulation for singular degenerate SPDEs, with proven convergence and practical demonstration.

Findings

Convergence of the numerical scheme to the unique solution.

Implementation of a finite element method for spatial discretization.

Numerical simulations validating the scheme's practicality.

Abstract

We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which 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.