Details
| Original language | English |
|---|---|
| Pages (from-to) | 529-561 |
| Number of pages | 33 |
| Journal | Numerische Mathematik |
| Volume | 139 |
| Issue number | 3 |
| Early online date | 6 Mar 2018 |
| Publication status | Published - Jul 2018 |
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.
ASJC Scopus subject areas
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Numerische Mathematik, Vol. 139, No. 3, 07.2018, p. 529-561.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Nonlinear discontinuous Petrov–Galerkin methods
AU - Carstensen, C.
AU - Bringmann, P.
AU - Hellwig, F.
AU - Wriggers, P.
N1 - 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 ‘Multiscale Problems: Algorithms, Numerical Analysis and Computation’. 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 ‘Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project ‘Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics’ (CA 151/22-1 and WR 19/51-1).
PY - 2018/7
Y1 - 2018/7
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85046747599&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1710.00529
DO - 10.48550/arXiv.1710.00529
M3 - Article
AN - SCOPUS:85046747599
VL - 139
SP - 529
EP - 561
JO - Numerische Mathematik
JF - Numerische Mathematik
SN - 0029-599X
IS - 3
ER -