In this paper, we aim to study the stability and convergence of a finite difference scheme for solving the two-dimensional nonlinear multi-term time fractional subdiffusion equation with weakly singular solutions. We apply the L1 scheme to discretize the multi-term temporal Caputo derivatives, a standard central difference method in space, and a backward formula to approximate the nonlinear term on the uniform mesh, respectively. Stability and pointwise-in-time error estimates are obtained for the fully discrete scheme. The global convergence order is ${\alpha }_1$, and the local convergence order is 1 in the temporal direction. The theoretical analysis is verified by some numerical results.

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.