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This work deals with the efficient numerical solution of the time-fractionalheat equation discretized on non-uniform temporal meshes. Non-uniform grids areessential to capture the singularities of "typical" solutions oftime-fractional problems. We propose an efficient space-time multigrid methodbased on the waveform relaxation technique, which accounts for the nonlocalcharacter of the fractional differential operator. To maintain an optimalcomplexity, which can be obtained for the case of uniform grids, we approximatethe coefficient matrix corresponding to the temporal discretization by itshierarchical matrix (${\cal H}$-matrix) representation. In particular, theproposed method has a computational cost of ${\cal O}(k N M \log(M))$, where$M$ is the number of time steps, $N$ is the number of spatial grid points, and$k$ is a parameter which controls the accuracy of the ${\cal H}$-matrixapproximation. The efficiency and the good convergence of the algorithm, whichcan be theoretically justified by a semi-algebraic mode analysis, aredemonstrated through numerical experiments in both one- and two-dimensionalspaces.

Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation