# Efficient solvers for the time-fractional heat equation based on multigrid waveform relaxation

## Meeting Details

Abstract: Fractional calculus has become increasingly popular in recent years due to their frequent appearance in various applications. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations where the coefficient matrix is dense. The standard solution of such systems requires a very high computational cost in addition to a high storage cost. This is quite different from the integer differential operators, which typically yield sparse coefficient matrices that can be efficiently solved by fast iterative methods with $O(n)$ complexity, being $n$ the number of grid points. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. In this work, we propose parallel-in-time multigrid algorithms based on the waveform relaxation approach, for uniform and non-uniform temporal grids. A semi-algebraic mode analysis is used to theoretically confirm the good convergence results obtained in all cases.