Multigrid Methods for Time-Fractional Evolution Equations:A Numerical Study

摘要

In this work,we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of orderα(0,1)in time.The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time.At each time step,instead of solving the linear algebraic system exactly,we employ a multigrid iteration with a Gauss–Seidel smoother to approximate the solution efficiently.Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.