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

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 [L. Chen, R. H. Nochetto, E. Ot{\'a}rola, A. J. Salgado, Math. Comp. 85 (2016) 2583--2607] is chosen to be the spatial solver. Motivated by [B. S. Southworth, SIAM J. Matrix Anal. Appl. 40 (2019) 564--608], 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 [X. Q. Yue, S. Shu, X. W. Xu, W. P. Bu, K. J. Pan, Comput. Math. Appl. 78 (2019) 3471--3484]. Numerical computations are included to confirm the theoretical predictions and demonstrate the sharpness of the derived convergence upper bound.