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

## TL;DR

This paper introduces an efficient multigrid waveform relaxation method utilizing hierarchical matrices for solving the time-fractional heat equation on non-uniform grids, achieving optimal complexity and good convergence.

## Contribution

It proposes a novel space-time multigrid approach with hierarchical matrix approximation to efficiently solve non-uniform grid discretizations of the time-fractional heat equation.

## Key findings

- Method achieves ${
m O}(k N M \log(M))$ complexity.
- Numerical experiments confirm efficiency and convergence.
- Applicable to 1D and 2D problems.

## Abstract

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.