Weak convergence of the L1 scheme for a stochastic subdiffusion problem driven by fractionally integrated additive noise

Abstract

The weak convergence of a fully discrete scheme for approximating a stochastic subdiffusion problem driven by fractionally integrated additive noise is studied. The Caputo fractional derivative is approximated by the L1 scheme and the Riemann-Liouville fractional integral is approximated with the first order convolution quadrature formula. The noise is discretized by using the Euler method and the spatial derivative is approximated with the linear finite element method. Based on the nonsmooth data error estimates of the corresponding deterministic problem, the weak convergence orders of the fully discrete schemes for approximating the stochastic subdiffusion problem driven by fractionally integrated additive noise are proved by using the Kolmogorov equation approach. Numerical experiments are given to show that the numerical results are consistent with the theoretical results.