Robust operator estimates and the application to substructuring methods for first-order systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, p. 1473-1494
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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.

DOI : https://doi.org/10.1051/m2an/2014006
Classification:  65N30
Keywords: first-order systems, Petrov-Galerkin methods, saddle point problems
@article{M2AN_2014__48_5_1473_0,
     author = {Wieners, Christian and Wohlmuth, Barbara},
     title = {Robust operator estimates and the application to substructuring methods for first-order systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     pages = {1473-1494},
     doi = {10.1051/m2an/2014006},
     mrnumber = {3264362},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_5_1473_0}
}
Wieners, Christian; 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, Volume 48 (2014) no. 5, pp. 1473-1494. doi : 10.1051/m2an/2014006. http://www.numdam.org/item/M2AN_2014__48_5_1473_0/

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