A spectrally accurate approximation to subdiffusion equations using the log orthogonal functions

In this paper, we develop and analyze a spectral-Galerkin method for solving subdiffusion equations, which contain Caputo fractional derivatives with order ν in (0, 1). The basis functions of our spectral method are constructed by applying a log mapping to Laguerre functions and have already been proved to be suitable to approximate functions with fractional power singularities in [S. Chen and J. Shen, Log Orthogonal Functions: Approximation Properties and Applications, preprint, arXiv:2003.01209[math.NA], 2020]. We provide rigorous regularity and error analysis which show that the scheme is spectrally accurate, i.e., the convergence rate depends only on regularity of problem data. The proof relies on the approximation properties of some reconstruction of the basis functions as well as the sharp regularity estimate in some weighted Sobolev spaces. Numerical experiments fully support the theoretical results and show the efficiency of the proposed spectral- Galerkin method. We also develop a fully discrete scheme with the proposed spectral method in time and the Galerkin finite element method in space, and apply the proposed techniques to subdiffusion equations with time-dependent diffusion coefficients as well as to the nonlinear time-fractional Allen-Cahn equation.