The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation

Based on L1 formula and the multiscale Galerkin method, a fully-discrete scheme is proposed for solving time fractional subdiffusion equations with α order Caputo fractional derivative. The existence and uniqueness of the solution of the fully-discrete scheme are proved, and the optimal convergence order O(h^r+ τ ^2-α) is also deduced, where r is the order of piecewise polynomials. A multilevel augmentation method (MAM) is developed to solve the linear systems resulting from the fully-discrete scheme at each time step, and MAM preserves the optimal convergence order. A numerical experiment is presented at last to show the validity of the theoretical analysis.