A hierarchical a posteriori error estimator for the Reduced Basis Method

Abstract

In this contribution, we are concerned with tight a posteriori error estimation for projection-based model order reduction of $\inf $-$\sup $ stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy, we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the $\inf $-$\sup $ constant may become small depending on the parameter. In such cases, a standard residual-based error estimator—complemented by the successive constrained method to compute a lower bound of the parameter dependent $\inf $-$\sup $ constant—may become infeasible.