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

TL;DR

This paper develops a multigrid-reduction-in-time (MGRIT) solver with a new two-level convergence analysis for unsteady fractional Laplacian problems, improving parallel efficiency and removing previous assumptions.

Contribution

It introduces a novel MGRIT algorithm with FCF-relaxation and a new temporal eigenvalue approximation, along with a generalized convergence theory that relaxes earlier assumptions.

Findings

The new MGRIT algorithm achieves faster convergence in numerical tests.

Theoretical convergence bounds are validated and shown to be sharp.

The method effectively handles unsteady fractional Laplacian problems with improved parallel performance.

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 [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.