Reconstruction-Based a Posteriori Error Estimates for the L1 Method for Time Fractional Parabolic Problems (Journal of Computational Mathematics, 2024)
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 theinitial singularity. To derive optimal order a posteriori error estimates, the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction forthe first-step integration are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretizationparameters and the data of the problems are derived. Various numerical experiments forthe one-dimensional linear fractional parabolic equations with smooth or nonsmooth exactsolution are used to verify and complement our theoretical results, with the convergenceof $α$ order for the nonsmooth case on a uniform mesh. To recover the optimal convergenceorder $2−α$ on a nonuniform mesh, we further develop a time adaptive algorithm by meansof barrier function recently introduced. The numerical implementations are performed onnonsmooth case again and verify that the true error and a posteriori error can achieve theoptimal convergence order in adaptive mesh.
Keywords: Time fractional parabolic differential equations; A posteriori error estimates; L1 method; Fractional integral reconstruction; Quadratic reconstruction.