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Description
A preconditioner for solving fractional partial differential equations (PDEs) is presented. In our method the fractional PDE is discretized on an adaptive grid, resulting in a Hierarchical matrix representation. The stiffness matrix has Toeplitz blocks along the diagonal and low-rank approximations off the diagonal. Our preconditioner expands on previously developed methods of conditioning Toeplitz systems with circulant matrices. We show how these methods can be applied cheaply on the adaptive mesh and prove that the spectrum of the resulting system is well-clustered. In order to prove these results, we must take special consideration of how the low-rank blocks perturb the eigenvalues of the Toeplitz block-diagonal system. We validate our results for various fractional orders and inspect any assumptions through numerical tests.