A Spectral Localization Method for Time-Fractional Integro-Differential Equations with Nonsmooth Data

In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal discretization, we employ the contour integral method (CIM) with parameterized hyperbolic contours to approximate the nonlocal operators. For spatial discretization, the standard piecewise linear Galerkin finite element method (FEM) is used. We then provide a rigorous error analysis, demonstrating that the proposed scheme achieves high accuracy even for problems with nonsmooth/vanishing initial values or low-regularity solutions, featuring spectral accuracy in time and second-order convergence in space. Finally, a series of numerical experiments in both 1-D and 2-D validate the theoretical findings and confirm that the algorithm combines the advantages of spectral accuracy, low computational cost, and efficient memory usage.