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2021 Convergence of multilevel spectral deferred corrections
Gitte Kremling, Robert Speck
Commun. Appl. Math. Comput. Sci. 16(2): 227-265 (2021). DOI: 10.2140/camcos.2021.16.227

Abstract

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.

Citation

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Gitte Kremling. Robert Speck. "Convergence of multilevel spectral deferred corrections." Commun. Appl. Math. Comput. Sci. 16 (2) 227 - 265, 2021. https://doi.org/10.2140/camcos.2021.16.227

Information

Received: 13 August 2020; Revised: 27 May 2021; Accepted: 17 June 2021; Published: 2021
First available in Project Euclid: 29 March 2022

Digital Object Identifier: 10.2140/camcos.2021.16.227

Subjects:
Primary: 65L20 , 65M12

Keywords: convergence theory , FAS , multilevel spectral deferred corrections , nonlinear multigrid , spectral deferred corrections

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.16 • No. 2 • 2021
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