Contractive local adaptive smoothing based on Dörfler's marking in a-posteriori-steered p-robust multigrid solvers

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00541717" target="_blank" >RIV/67985840:_____/21:00541717 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1515/cmam-2020-0024" target="_blank" >https://doi.org/10.1515/cmam-2020-0024</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1515/cmam-2020-0024" target="_blank" >10.1515/cmam-2020-0024</a>

Result language

angličtina

Original language name

Contractive local adaptive smoothing based on Dörfler's marking in a-posteriori-steered p-robust multigrid solvers

Original language description

n 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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA20-01074S" target="_blank" >GA20-01074S: Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2021

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Computational Methods in Applied Mathematics

ISSN

1609-4840

e-ISSN

1609-9389

Volume of the periodical

21

Issue of the periodical within the volume

2

Country of publishing house

DE - GERMANY

Number of pages

24

Pages from-to

445-468

UT code for WoS article

000634948900011

EID of the result in the Scopus database

2-s2.0-85100534705