Reconstruction-Based a Posteriori Error Estimates for the L1 Method for Time Fractional Parabolic Problems

Abstract

In this paper, we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations, whose solutions have generally the initial singularity. To derive optimal order a posteriori error estimates, the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived. Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results, with the convergence of $α$ order for the nonsmooth case on a uniform mesh. To recover the optimal convergence order $2−α$ on a nonuniform mesh, we further develop a time adaptive algorithm by means of barrier function recently introduced. The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.