A stabilized scheme satisfying the discrete maximum principle for a time fractional convection-diffusion-reaction equation

We study a time fractional convection-diffusion-reaction equation in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$. A stabilized numerical scheme satisfying a discrete maximum principle is constructed by combining the conforming linear finite element method with the algebraic flux correction method. The resulting semi-discrete scheme is nonlinear, and its well-posedness is established. Assuming nonsmooth initial data, we derive error estimates for the semi-discrete scheme using energy arguments. For the temporal discretization, we employ the L1 method, obtaining a fully discrete scheme for which we prove well-posedness and the discrete maximum principle. We also present numerical experiments that validate the order of convergence as well as we test our schemes to solutions that possess layers.