Algebraic multilevel preconditioning in spectral fractional diffusion

Abstract

The numerical solution of linear systems obtained as a result of discretization of a spectral fractional diffusion problem is studied. The finite element method is applied to the considered boundary value problem. The system matrix is a fractional power of the product of the inverse of the mass matrix and the stiffness matrix. The matrix thus defined is symmetric and positive definite (SPD) with respect to the inner product associated with the mass matrix, but is dense, which is consistent with the nonlocal nature of fractional diffusion. The presented results are in the spirit of the BURA (Best Uniform Rational Approximation) method. BURA reduces numerical solution of the dense linear system to the solution of k systems with sparse SPD diffusion-reaction matrices, where k is the degree of rational approximation. We prove the existence of algebraic multilevel iteration (AMLI) methods for preconditioning such type of emergent matrices that satisfy the conditions for optimal computational complexity. Both multiplicative and additive AMLI preconditioners have been developed, determining the minimum possible degree θ of the hierarchical θ-refinement of the mesh.