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Robust operator estimates and the application to substructuring methods 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 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.

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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