A fast locally refined method for the boundary value problem of fractional diffusion equations

We develop a finite volume method on a geometrically refined mesh to effectively treat the boundary layer singularity of the solutions to the boundary value problem of fractional differential equations. We further exploit the structure of the stiffness matrix to develop a properly modified fast conjugate gradient method for the efficient solution of the linear algebraic system. The fast method has a computational work count of O (N log
²
N) per iteration and a memory requirement of O (N) where N is the number of grid point, despite that the stiffness matrix is a full matrix. Numerical results are presented to demonstrate the strong potential of the method.