A parallel-in-time multigrid solver with a new two-level convergence for two-dimensional 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 an MGRIT algorithm, using FCF-relaxation with time-dependent time-grid propagators, to seek the finite element approximations of two-dimensional unsteady fractional Laplacian problems. Motivated by [B. S. Southworth, SIAM J. Matrix Anal. Appl. 40 (2019), pp. 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. Numerical computations are included to confirm theoretical predictions and demonstrate the sharpness of the derived convergence upper bound.