Robust operator estimates and the application to substructuring methods for first-order systems

Christian Wieners; Barbara Wohlmuth

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 5, page 1473-1494
  • ISSN: 0764-583X

Abstract

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We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz et al., SIAM J. Numer. Anal. 49 (2011) 1788–1809; Zitelli et al., J. Comput. Phys. 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space. Here, we show that the abstract framework of saddle point problems and domain decomposition techniques provide stability and a priori estimates. To obtain efficient numerical algorithms, we use a second Schur complement reduction applied to the trial space. This restricts the degrees of freedom to the skeleton. We construct a preconditioner for the skeleton problem, and the efficiency of the discretization and the solution method is demonstrated by numerical examples.

How to cite

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Wieners, Christian, and Wohlmuth, Barbara. "Robust operator estimates and the application to substructuring methods for first-order systems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1473-1494. <http://eudml.org/doc/273259>.

@article{Wieners2014,
abstract = {We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz et al., SIAM J. Numer. Anal. 49 (2011) 1788–1809; Zitelli et al., J. Comput. Phys. 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space. Here, we show that the abstract framework of saddle point problems and domain decomposition techniques provide stability and a priori estimates. To obtain efficient numerical algorithms, we use a second Schur complement reduction applied to the trial space. This restricts the degrees of freedom to the skeleton. We construct a preconditioner for the skeleton problem, and the efficiency of the discretization and the solution method is demonstrated by numerical examples.},
author = {Wieners, Christian, Wohlmuth, Barbara},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {first-order systems; Petrov–Galerkin methods; saddle point problems; substructuring methods; discontinuous Petrov-Galerkin method; preconditioning},
language = {eng},
number = {5},
pages = {1473-1494},
publisher = {EDP-Sciences},
title = {Robust operator estimates and the application to substructuring methods for first-order systems},
url = {http://eudml.org/doc/273259},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Wieners, Christian
AU - Wohlmuth, Barbara
TI - Robust operator estimates and the application to substructuring methods for first-order systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1473
EP - 1494
AB - We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz et al., SIAM J. Numer. Anal. 49 (2011) 1788–1809; Zitelli et al., J. Comput. Phys. 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space. Here, we show that the abstract framework of saddle point problems and domain decomposition techniques provide stability and a priori estimates. To obtain efficient numerical algorithms, we use a second Schur complement reduction applied to the trial space. This restricts the degrees of freedom to the skeleton. We construct a preconditioner for the skeleton problem, and the efficiency of the discretization and the solution method is demonstrated by numerical examples.
LA - eng
KW - first-order systems; Petrov–Galerkin methods; saddle point problems; substructuring methods; discontinuous Petrov-Galerkin method; preconditioning
UR - http://eudml.org/doc/273259
ER -

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