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

This work deals with the efficient numerical solution of the time-fractional heat equation discretized on non-uniform temporal meshes. Non-uniform grids are essential to capture the singularities of "typical" solutions of time-fractional problems. We propose an efficient space-time multigrid method based on the waveform relaxation technique, which accounts for the nonlocal character of the fractional differential operator. To maintain an optimal complexity, which can be obtained for the case of uniform grids, we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix (${\cal H}$-matrix) representation. In particular, the proposed 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 ${\cal H}$-matrix approximation. The efficiency and the good convergence of the algorithm, which can be theoretically justified by a semi-algebraic mode analysis, are demonstrated through numerical experiments in both one- and two-dimensional spaces.