Finite element analysis of a multi-term nonlinear time-fractional convection-diffusion equation with Caputo-Fabrizio derivative

This paper presents a novel numerical framework for solving a two-dimensional multi-term nonlinear convection-diffusion equation involving the Caputo-Fabrizio time-fractional derivative. The proposed method begins with a semi-discrete formulation based on a spatially continuous finite element method, and an optimal spatial error estimate of order $O(h^{2})$ for the semi-discrete scheme is rigorously established. To advance the solution in time, a fully discrete scheme is constructed by integrating a second-order finite difference discretization, ensuring compatibility with the nonlinear terms. The resulting algorithm is proven to be unconditionally stable in the $L^{2} $ -norm. Furthermore, under suitable regularity assumptions, the method attains a convergence rate of $O(h^{2} + \overline{\tau}^{2}) $ , where h and τ̅ represent the spatial and temporal discretization steps, respectively.