Adaptive fast L1 − 2 scheme for solving time fractional parabolic problems

In this paper, we study a posteriori error estimates of the fast $L1-2$ scheme for time discretization of time fractional parabolic differential equations. To overcome the huge workload caused by the nonlocality of fractional derivative, a fast algorithm is applied to the construction of the $L1-2$ scheme. Employing the numerical solution obtained by the fast $L1-2$ scheme, a piecewise continuous function approximating the exact solution is constructed. Then, by exploring the error equations, a posteriori error estimates are obtained in different norms, which depend only on the discretization parameters and the data of the problems. Various numerical experiments for the fractional parabolic equations with smooth or nonsmooth exact solutions on different time meshes, including the frequently-used graded mesh, are carried out to verify and complement our theoretical results. Based on the obtained a posteriori error estimates, a time adaptive algorithm is proposed to reduce the computational cost substantially and provides efficient error control for the solution.