L1 scheme for semilinear stochastic subdiffusion with integrated fractional Gaussian noise

Abstract

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter H∈(1/2,1). The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order α∈(0,1) and the Riemann–Liouville time-fractional integral of order γ∈(0,1), respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order O(τmin{H+α+γ−1−ε,α}),ε>0. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method.