Volume 48, Number 5, September-October 2014
|Page(s)||1473 - 1494|
|Published online||13 August 2014|
Robust operator estimates and the application to substructuring methods for first-order systems
1 Institut für Angewandte und
Numerische Mathematik, KIT, Karlsruhe, Germany.
2 Fakultät Mathematik M2, Technische Universität München, Garching, Germany.
Revised: 12 January 2014
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.
Mathematics Subject Classification: 65N30
Key words: First-order systems / Petrov–Galerkin methods / saddle point problems
© EDP Sciences, SMAI 2014
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