Geodesic
Robust operator estimates and the application to substructuring methods for first-order systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 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 : 10.1051/m2an/2014006
Classification : 65N30
Keywords: first-order systems, Petrov-Galerkin methods, saddle point problems 
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     title = {Robust operator estimates and the application to substructuring methods for first-order systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1473--1494},
     publisher = {EDP-Sciences},
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     year = {2014},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014006/}
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Wieners, Christian; Wohlmuth, Barbara. Robust operator estimates and the application to substructuring methods for first-order systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1473-1494. doi: 10.1051/m2an/2014006

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