A Parallel-in-time Multigrid Solver For Two-dimensional Unsteady Fractional Laplacian Problems

Time-dependent problems involving fractional Laplacian operator are of particular interest in fractional calculus.One of the main challenges lies in the nonlocality of the fractional Laplacian.A widely used approach is to utilize the Caffarelli-Silvestre extension to reformulate the original problem into some local problem,however,incorporating one more dimension in space.Another difficulty is how to design an efficient space-time parallel solver.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 study 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?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(e.g.,error-norm decay rate in ???is about DoF^-1/3and the multigrid V?1,1?-cycle with line smoother proposed in [L.Chen,R.H.Nochetto,E.Ot?arola and A.J.Salgado,Math.Comput.85?2016?2583-2607] is almost uniform with respect to the fractional order and Do F or,equivalently,the spatial mesh size,and demonstrate the sharpness of the derived convergence upper bound.