A Hierarchical A-Posteriori Error Estimatorfor the Reduced Basis Method

Hain S, Ohlberger M, Radic M, Urban K

Forschungsartikel (Zeitschrift)

Zusammenfassung

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 weak 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.

Details zur Publikation

Veröffentlichungsjahr: 2019
Sprache, in der die Publikation verfasst ist: Englisch
Link zum Volltext: http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f6vAoGVkQUY3HEifyt4P98-3D_zWyY9QRoJEuEfyxxv1fIX3dKPGcuY4hNT6y7Qsm-2Bv14ihCC2EFl3ZInJiwhsn0PaQlyPxaEePT1Znuga-2FHGSgW48mznzW-2B-2FOxwUcsXerUcgFIelOQ4PFQjg5rNBX-2FQ0lTcHHDoR30XajLnBpuCJyGDBBWy3Qjpwjl-2BLxZb