In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $L1$-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.

In this paper, a fully discrete finite volume element (FVE) scheme is proposed to treat the linear multi-term time fractional reaction-diffusion models with variable coefficients on triangular grids, where the Caputo fractional derivative of time variable is approximated by the $L1$ formula. The existence and uniqueness of solution for the proposed scheme is proved based on the matrix theories, the stability results are derived in detail by using a multi-term fractional Gronwall inequality, and the optimal error estimates under the $L^2(\mathrm{\Omega })$ and $H^1(\mathrm{\Omega })$ norms are also obtained. Owing to the variable coefficients in the models, a special technique should be used to analyze the stability and error estimates. Finally, three numerical examples with different fractional derivative terms are given to validate the feasibility and effectiveness.