Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation

A two-dimensional multi-term time fractional diffusion equation $D_t^{\mathit{\alpha }}u(x,y,t)-\mathrm{\Delta }u(x,y,t)=f(x,y,t)$ is considered in this paper, where $D_t^{\mathit{\alpha }}$ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.