A-posteriori error estimations of the GJF-Petrov–Galerkin methods for fractional differential equations

Abstract

In this paper, we develop an efficient spectral GJF-Petrov–Galerkin algorithm and the postprocessed method to solve a class of fractional initial value problems. The main part of these algorithms is to use a special set of general Jacobi functions (GJFs) to form the trial space and test space. We give a rigorous error analysis in non-uniformly weighted Sobolev spaces and obtain optimal error estimates. In particular, the postprocessing technique is used to construct the postprocessed method. In addition, we derive its superconvergence estimates and define a-posteriori error estimators that are asymptotically exact. Numerical experiments are included to support the theoretical analysis.