Geodesic
A discontinuous Galerkin reduced basis element method for elliptic problems
Antonietti, Paola F.1  ; Pacciarini, Paolo1 ; Quarteroni, Alfio1, 2 
1 MOX–Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
2 CMCS, École Polytechnique Fédérale de Lausanne (EPFL), Station 8, 1015 Lausanne, Switzerland.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 337-360

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We propose and analyse a new discontinuous reduced basis element method for the approximation of parametrized elliptic PDEs in partitioned domains. The method is built upon an offline stage (parameter independent) and an online (parameter dependent) one. In the offline stage we build a non-conforming (discontinuous) global reduced space as a direct sum of local basis functions generated independently on each subdomain. In the online stage, for any given value of the parameter, the approximate solution is obtained by ensuring the weak continuity of the fluxes and of the solution itself thanks to a discontinuous Galerkin approach. The new method extends and generalizes the methods introduced in [L. Iapichino, Ph.D. thesis, EPF Lausanne (2012); L. Iapichino, A. Quarteroni and G. Rozza, Comput. Methods Appl. Mech. Eng. 221–222 (2012) 63–82]. We prove its stability and convergence properties, as well as the spectral properties of the associated online algebraic system. We also propose a two-level preconditioner for the online problem which exploits the pre-existing decomposition of the domain and is based upon the introduction of a global coarse finite element space. Numerical tests are performed to verify our theoretical results.

Reçu le :
DOI : 10.1051/m2an/2015045
Classification : 65N12, 65N30
Keywords: Reduced basis element method, discontinuous Galerkin, domain decomposition

Antonietti, Paola F. 1 ; Pacciarini, Paolo 1 ; Quarteroni, Alfio 1, 2

1 MOX–Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
2 CMCS, École Polytechnique Fédérale de Lausanne (EPFL), Station 8, 1015 Lausanne, Switzerland. 
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     title = {A discontinuous {Galerkin} reduced basis element method for elliptic problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {337--360},
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Antonietti, Paola F.; Pacciarini, Paolo; Quarteroni, Alfio. A discontinuous Galerkin reduced basis element method for elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 337-360. doi: 10.1051/m2an/2015045

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