On Spectral Petrov–Galerkin Method for Solving Time-Fractional Diffusion Equations in Weighted Sobolev Space

Source abstract

Time-fractional diffusion equations with a Caputo derivative of order α∈(0,1) provide effective models for anomalous diffusion processes with memory effects, while the weakly singular fractional kernel and rough source terms generally lead to limited temporal regularity. In this paper, we develop a fully discrete scheme that combines a second-order central difference approximation in space with a spectral Petrov–Galerkin method in time. The spatial semidiscretization leads to fractional initial value problems with reaction parameters given by the eigenvalues of the spatial discretization matrix. Unlike the standard reaction-free case, the presence of these reaction parameters complicates the weighted regularity analysis. An ultraweak formulation and a bootstrapping argument are employed to establish the temporal regularity of the semidiscrete solution in weighted Sobolev spaces for source terms with regularity index r≥−α, including rough inputs. The dependence of the resulting regularity estimates on the reaction coefficients is explicitly characterized for the subsequent fully discrete error analysis. Under suitable spatial regularity of the source term, the fully discrete error is bounded by C(h2+N−m), where m=α+min{r,2α+1−ϵ} and ϵ>0 is an arbitrarily small positive constant. No coupling condition between the spatial mesh size h and the temporal polynomial degree N is required. A fast iterative solver is also developed using fast polynomial transforms and matrix–vector products. Numerical experiments support the theoretical convergence rates. For the tested scalar FIVP, the fast solver requires less computational time than the direct solver at the larger temporal polynomial degrees.