hp-FEM for the α-Mosolov problem

a priori and a posteriori error estimates

authored by: Lothar Banz, Ernst P. Stephan
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.
Organisation(s): Institute of Applied Mathematics
External Organisation(s): University of Salzburg
Type: Article
Journal: Numerische Mathematik
Volume: 156
Pages: 1679-1718
No. of pages: 40
ISSN: 0029-599X
Publication date: 10.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.1007/s00211-024-01433-8 (Access: Open)

BibTeX

@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",
}