Nonlinear discontinuous Petrov–Galerkin methods

authored by: C. Carstensen, P. Bringmann, F. Hellwig, P. Wriggers
Abstract: The discontinuous Petrov–Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Humboldt-Universität zu Berlin (HU Berlin)
Type: Article
Journal: Numerische Mathematik
Volume: 139
Pages: 529-561
No. of pages: 33
ISSN: 0029-599X
Publication date: 07.2018
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.48550/arXiv.1710.00529 (Access: Open)
https://doi.org/10.1007/s00211-018-0947-5 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{0201a557ff0a449e803869da09ffac1a,
title = "Nonlinear discontinuous Petrov–Galerkin methods",
abstract = "The discontinuous Petrov–Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples.",
author = "C. Carstensen and P. Bringmann and F. Hellwig and P. Wriggers",
note = "Funding information: The work has been written while the first author enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program {\textquoteleft}Multiscale Problems: Algorithms, Numerical Analysis and Computation{\textquoteright}. The second and third author were supported by the Berlin Mathematical School. The research of all four authors has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 {\textquoteleft}Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis{\textquoteright} under the project {\textquoteleft}Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics{\textquoteright} (CA 151/22-1 and WR 19/51-1).",
year = "2018",
month = jul,
doi = "10.48550/arXiv.1710.00529",
language = "English",
volume = "139",
pages = "529--561",
journal = "Numerische Mathematik",
issn = "0029-599X",
publisher = "Springer New York",
number = "3",
}