Numerical study for a stochastic semilinear subdiffusion equation driven by fractional Brownian motions

Abstract

In this work, we consider the Galerkin finite element method for solving the stochastic semilinear subdiffusion equation driven by additive fractional Brownian motion with Hurst parameter H ∈ ( 0 , 1 ) , where the fractional Brownian motion has a Wiener integration representation. The existence and uniqueness of a mild solution are proved using the Banach fixed point theorem. The temporal and spatial regularity of the solution are studied via the semigroup approach. The finite element method is used to approximate the spatial variable. The Caputo fractional time derivative and the Riemann–Liouville integral are approximated using the Grünwald–Letnikov schemes, respectively, and the noise is discretized using the Euler method. The optimal error estimates of the fully discrete scheme are established using the discrete Laplace transform method. Under the assumption that the noise is in the trace class, we prove that the time convergence order is O ( τ min { α , H + α + γ − 1 , 1 / 2 } ) when H ∈ ( 0 , 1 / 2 ) , and O ( τ min { α , H + α + γ − 1 } ) when H ∈ ( 1 / 2 , 1 ) . Here, τ denotes the time step size. Numerical experiments are conducted to validate the theoretical results.