Oberseminar "Aktuelle Themen der Numerik"
Donnerstag, 11.07.2013, 14.00 Uhr
A Reduced basis method for convection-dominated problems
Dr. Gerrit Welper (RWTH Aachen)
Reduced basis methods for parametric transport and convection-dominated problems face two serious issues: The known stabilization methods typically do not produce (up to constants) optimal solutions and the corresponding norms do not allow a tight relation between error and residual. However, these two ingredients are crucial for the construction of surrogates that lead to rate optimal greedy methods for the construction of the reduced basis as well as the online solution and certification.
Therefore, we develop well-conditioned variational formulations for transport and convection-diffusion problems that allow a tight relation between error and residual. In fact, by a proper choice of the variational formulation these constants can be one. Based on that formulation, we then obtain stabilization methods that yield (up to constants) optimal approximations of the solution. The stability of these methods relies on an inf-sup constant which can, in principle, be driven as close to one as one wishes. Since the stabilization method does not use any structural properties of the basis, it can be applied to both the truth and reduced spaces. For the latter one, the inf-sup stability is ensured by an additional inner greedy loop. In addition, standard offline/online decompositions are applicable, so that we arrive at a reduced basis method yielding rate optimal reduced bases as well as rigorous error certification for convection-diffusion problems.
Zeit: 14:00 Uhr
Ort: Raum 149, Hauptgebäude, RWTH Aachen, Templergraben 55, 52056 Aachen