A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems

Abstract

The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm using FCF-relaxation with time-dependent time-grid propagators to seek the finite element approximations of unsteady fractional Laplacian problems. The multigrid with line smoother proposed in Chen et al. (2016) is chosen to be the spatial solver. Motivated by Southworth (2019), we provide a new temporal eigenvalue approximation property and then deduce a generalized two-level convergence theory which removes the previous unitary diagonalization assumption on the fine and coarse time-grid propagators required in Yue et al. (2019). Numerical computations are included to confirm the theoretical predictions and demonstrate the sharpness of the derived convergence upper bound.