A Discontinuous Galerkin Reduced Basis Element Method For Elliptic Problems

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.