Vol. 16, No. 2, 2021

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Convergence of multilevel spectral deferred corrections

Gitte Kremling and Robert Speck

Vol. 16 (2021), No. 2, 227–265

The spectral deferred correction (SDC) method is a class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well known, for suitable problems it gains one order per iteration up to the order of the quadrature method of the collocation problem provided. This appealing feature enables an easy creation of flexible, high-order accurate methods for ODEs. A variation of SDC are multilevel spectral deferred corrections (MLSDC). Here, iterations are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. While there are several numerical examples which show its capabilities and efficiency, a theoretical convergence proof is still missing. We address this issue. A proof of the convergence of MLSDC, including the determination of the convergence rate in the time-step size, will be given and the results of the theoretical analysis will be numerically demonstrated. It turns out that there are restrictions for the advantages of this method over SDC regarding the convergence rate.

spectral deferred corrections, multilevel spectral deferred corrections, convergence theory, nonlinear multigrid, FAS
Mathematical Subject Classification
Primary: 65L20, 65M12
Received: 13 August 2020
Revised: 27 May 2021
Accepted: 17 June 2021
Published: 2 November 2021
Gitte Kremling
Juelich Supercomputing Centre
Forschungszentrum Juelich GmbH
Robert Speck
Juelich Supercomputing Centre
Forschungszentrum Juelich GmbH