Numerical Approximation for Stochastic Nonlinear Fractional Diffusion Equation Driven by Rough Noise

TL;DR

This paper develops a fully discrete numerical scheme for stochastic fractional diffusion equations driven by rough noise, combining regularization, finite element, and convolution quadrature methods, with theoretical error analysis and numerical validation.

Contribution

It introduces a novel combination of regularization and discretization techniques for stochastic fractional PDEs driven by rough noise, with comprehensive convergence analysis.

Findings

The regularity of solutions is established.

Convergence of the numerical scheme is proven.

Numerical examples confirm theoretical results.

Abstract

In this work, we are interested in building the fully discrete scheme for stochastic fractional diffusion equation driven by fractional Brownian sheet which is temporally and spatially fractional with Hurst parameters $H_{1}, H_{2} \in(0,\frac{1}{2}]$. We first provide the regularity of the solution. Then we employ the Wong-Zakai approximation to regularize the rough noise and discuss the convergence of the approximation. Next, the finite element and backward Euler convolution quadrature methods are used to discretize spatial and temporal operators for the obtained regularized equation, and the detailed error analyses are developed. Finally, some numerical examples are presented to confirm the theory.