Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers

Ani Miraçi; Jan Papež; Martin Vohralík

doi:10.1515/cmam-2020-0024

Published by De Gruyter February 5, 2021

Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers

Ani Miraçi , Jan Papež and Martin Vohralík

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2020-0024

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Abstract

In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p-robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 . After one V-cycle (“full-smoothing” substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (“adaptive-smoothing” substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.

Keywords: Finite Element Method; Multigrid Method; Schwarz Method; Block–Jacobi Smoother; A Posteriori Estimate; Adaptive Smoothing; Stable Decomposition

MSC 2010: 65N55; 65N30; 65F10; 65N15; 65N22

Funding source: European Research Council

Award Identifier / Grant number: 647134 GATIPOR

Funding statement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 647134 GATIPOR). The work of J. Papež was supported by the Grant Agency of the Czech Republic under Grant No. 20-01074S. The authors are grateful to Inria Sophia Antipolis – Méditerranée “NEF” computation cluster for providing resources and support.

Acknowledgements

We would like to thank the anonymous referees for their useful suggestions concerning this manuscript.

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Received: 2020-02-28

Revised: 2020-09-04

Accepted: 2020-12-16

Published Online: 2021-02-05

Published in Print: 2021-04-01

Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers

Abstract

Acknowledgements

References

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