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

Published online by Cambridge University Press:  13 August 2014

Christian Wieners  and
Barbara Wohlmuth
Christian Wieners
Affiliation:
Institut für Angewandte und Numerische Mathematik, KIT, Karlsruhe, Germany. . christian.wieners@kit.edu
Barbara Wohlmuth
Affiliation:
Fakultät Mathematik M2, Technische Universität München, Garching, Germany. ; wohlmuth@ma.tum.de
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Abstract

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite generalpartial differential operators. The starting point of our analysis is the DPG methodintroduced by [Demkowicz et al., SIAM J. Numer. Anal.49 (2011) 1788–1809; Zitelli et al., J.Comput. Phys. 230 (2011) 2406–2432]. This discretization resultsin a sparse positive definite linear algebraic system which can be obtained from a saddlepoint 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 domaindecomposition techniques provide stability and a priori estimates. Toobtain efficient numerical algorithms, we use a second Schur complement reduction appliedto the trial space. This restricts the degrees of freedom to the skeleton. We construct apreconditioner for the skeleton problem, and the efficiency of the discretization and thesolution method is demonstrated by numerical examples.

Keywords

First-order systemsPetrov–Galerkin methodssaddle point problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1473 - 1494
Copyright
© EDP Sciences, SMAI 2014

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