hp-FEM for the α-Mosolov problem: a priori and a posteriori error estimates

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Lothar Banz
  • Ernst P. Stephan

Organisationseinheiten

Externe Organisationen

  • Universität Salzburg
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Seiten (von - bis)1679-1718
Seitenumfang40
FachzeitschriftNumerische Mathematik
Jahrgang156
Ausgabenummer5
Frühes Online-Datum25 Sept. 2024
PublikationsstatusVeröffentlicht - Okt. 2024

Abstract

An hp-finite element discretization for the α-Mosolov problem, a scalar variant of the Bingham flow problem but with the α-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞) we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2. Numerical results underline our theoretical findings.

ASJC Scopus Sachgebiete

Zitieren

hp-FEM for the α-Mosolov problem: a priori and a posteriori error estimates. / Banz, Lothar; Stephan, Ernst P.
in: Numerische Mathematik, Jahrgang 156, Nr. 5, 10.2024, S. 1679-1718.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Banz L, Stephan EP. hp-FEM for the α-Mosolov problem: a priori and a posteriori error estimates. Numerische Mathematik. 2024 Okt;156(5):1679-1718. Epub 2024 Sep 25. doi: 10.1007/s00211-024-01433-8
Banz, Lothar ; Stephan, Ernst P. / hp-FEM for the α-Mosolov problem : a priori and a posteriori error estimates. in: Numerische Mathematik. 2024 ; Jahrgang 156, Nr. 5. S. 1679-1718.
Download
@article{031b56308da247c9b261d0d76504bf18,
title = "hp-FEM for the α-Mosolov problem: a priori and a posteriori error estimates",
abstract = "An hp-finite element discretization for the α-Mosolov problem, a scalar variant of the Bingham flow problem but with the α-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞) we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2. Numerical results underline our theoretical findings.",
keywords = "65K15, 65N30, 65N50",
author = "Lothar Banz and Stephan, {Ernst P.}",
note = "Publisher Copyright: {\textcopyright} The Author(s) 2024.",
year = "2024",
month = oct,
doi = "10.1007/s00211-024-01433-8",
language = "English",
volume = "156",
pages = "1679--1718",
journal = "Numerische Mathematik",
issn = "0029-599X",
publisher = "Springer New York",
number = "5",

}

Download

TY - JOUR

T1 - hp-FEM for the α-Mosolov problem

T2 - a priori and a posteriori error estimates

AU - Banz, Lothar

AU - Stephan, Ernst P.

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/10

Y1 - 2024/10

N2 - An hp-finite element discretization for the α-Mosolov problem, a scalar variant of the Bingham flow problem but with the α-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞) we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2. Numerical results underline our theoretical findings.

AB - An hp-finite element discretization for the α-Mosolov problem, a scalar variant of the Bingham flow problem but with the α-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞) we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2. Numerical results underline our theoretical findings.

KW - 65K15

KW - 65N30

KW - 65N50

UR - http://www.scopus.com/inward/record.url?scp=85204783311&partnerID=8YFLogxK

U2 - 10.1007/s00211-024-01433-8

DO - 10.1007/s00211-024-01433-8

M3 - Article

AN - SCOPUS:85204783311

VL - 156

SP - 1679

EP - 1718

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 5

ER -