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We develop a fast solver for the spectral element method (SEM) applied to thetwo-sided fractional diffusion equation on uniform, geometric and gradedmeshes. By approximating the singular kernel with a degenerate kernel, weconstruct a hierarchical matrix (H-matrix) to represent the stiffness matrix ofthe SEM and provide error estimates verified numerically. We can solveefficiently the H-matrix approximation problem using a hierarchical LUdecomposition method, which reduces the computational cost to $O(R^2 N_d\log^2N) +O(R^3 N_d \log N)$, where $R$ it is the rank of submatrices of theH-matrix approximation, $N_d$ is the total number of degrees of freedom and $N$is the number of elements. However, we lose the high accuracy of the SEM. Thus,we solve the corresponding preconditioned system by using the H-matrixapproximation problem as a preconditioner, recovering the high order accuracyof the SEM. The condition number of the preconditioned system is independent ofthe polynomial degree $P$ and grows with the number of elements, but at modestvalues of the rank $R$ is below order 10 in our experiments, which represents areduction of more than 11 orders of magnitude from the unpreconditioned system;this reduction is higher in the two-sided fractional derivative compared toone-sided fractional derivative. The corresponding cost is $O(R^2 N_d \log^2N)+O(R^3 N_d \log N)+O(N_d^2)$. Moreover, by using a structured mesh (uniformor geometric mesh), we can further reduce the computational cost to $O(R^2N_d\log^2 N) +O(R^3 N_d \log N)+ O(P^2 N\log N)$ for the preconditioned system.We present several numerical tests to illustrate the proposed algorithm using$h$ and $p$ refinements.

A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation