Fast solver for fractional differential equations based on Hierarchical Matrices

A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on nonuniform meshes. It is based on the utilization of hierarchical matrices (H-Matrices) for the representation of the stiffness matrix resulting from the finite element discretization of the FDEs. We employ a geometric multigrid method for the solution of the algebraic system of equations. We combine it with an adaptive algorithm based on a posteriori error estimation to deal with general-type singularities arising in the solution of the FDEs. Through various test examples we demonstrate the efficiency of the method and the highaccuracy of the numerical solution even in the presence of singularities. To the best of our knowledge, there are currently no other methods for FDEs that resolve singularities accurately at linear complexity as the one we propose here.