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.
DOI: 10.1051/m2an/2015045
Keywords: Reduced basis element method, discontinuous Galerkin, domain decomposition
@article{M2AN_2016__50_2_337_0, author = {Antonietti, Paola F. and Pacciarini, Paolo and Quarteroni, Alfio}, title = {A discontinuous {Galerkin} reduced basis element method for elliptic problems}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {337--360}, publisher = {EDP-Sciences}, volume = {50}, number = {2}, year = {2016}, doi = {10.1051/m2an/2015045}, mrnumber = {3482546}, zbl = {1343.65132}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015045/} }
TY - JOUR AU - Antonietti, Paola F. AU - Pacciarini, Paolo AU - Quarteroni, Alfio TI - A discontinuous Galerkin reduced basis element method for elliptic problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 337 EP - 360 VL - 50 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015045/ DO - 10.1051/m2an/2015045 LA - en ID - M2AN_2016__50_2_337_0 ER -
%0 Journal Article %A Antonietti, Paola F. %A Pacciarini, Paolo %A Quarteroni, Alfio %T A discontinuous Galerkin reduced basis element method for elliptic problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 337-360 %V 50 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015045/ %R 10.1051/m2an/2015045 %G en %F M2AN_2016__50_2_337_0
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