A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative

This paper focuses on some mathematical and numerical aspects of reaction-diffusion problems pertaining to non-integer time derivatives using the well-known method of lower and upper solutions combined with the monotone iterative technique. First, we study the existence and uniqueness of weak solutions of the proposed models, then we prove some comparison results. Besides, linear finite element spaces on triangles are used to discretize the problem in space, whereas the generalized backward-Euler method is adopted to approximate the time non-integer derivative. Furthermore, the idea of this method is to construct two sequences of solutions of a linear initial value problem which are easier to compute and converge to the solution of the nonlinear problem. We show numerically through two examples that this convergence requires only few iterations. Some well-known examples with exact solutions and numerical results based on the finite element method in 2D are provided to validate the theoretical results. As a result, we confirm that the proposed method is efficient and easy to use to overcome the convergence and stability difficulties.