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The multigrid-reduction-in-time (MGRIT) technique has proven to be successfulin achieving higher run-time speedup by exploiting parallelism in time. Thegoal of this article is to develop and analyze a MGRIT algorithm, usingFCF-relaxation with time-dependent time-grid propagators, to seek the finiteelement approximations of unsteady fractional Laplacian problems. The multigridwith 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 ageneralized two-level convergence theory which removes the previous unitarydiagonalization assumption on the fine and coarse time-grid propagatorsrequired 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 thetheoretical predictions and demonstrate the sharpness of the derivedconvergence upper bound.

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