On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

Modelling and Scientific Computing, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Station 8, EPFL, 1015 Lausanne, Switzerland
toni.lassila@epfl.ch; andrea.manzoni@epfl.ch; gianluigi.rozza@epfl.ch

Received: 9 August 2011
Revised: 30 March 2012

Abstract

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper correction term. Because the approximate problems are affine, an efficient offline/online computational scheme can be developed for the certified solution (error bounds and stability factors) of the parametric equations considered. We experiment with different correction terms suited for a posteriori error estimation of the reduced basis solution of elliptic coercive and noncoercive problems.