Reduced Basis Methods and A Posteriori Error Estimation for Incompressible Fluid Flows in Parametrized Domains
The reduced basis (RB) method is a model order reduction approach that permits the efficient yet reliable approximation of input-output relationships induced by parametrized partial differential equations. As the method recognizes, the solutions to a parametrized partial differential equation (for different parameter values) are not arbitrary members of the infinite-dimensional solution space, but rather reside or evolve on a much lower-dimensional manifold. Exploitation of this low-dimensionality is the key idea of the RB approach.
Although there are many existing examples of reduced order models for the Stokes and incompressible Navier-Stokes equations, only the RB method is endowed with practicable and rigorous a posteriori error bounds. Indeed, the development of rigorous RB error estimators presents the main methodological challenge, and has been focus of recent research.
The method is well-developed for linear elliptic and parabolic partial differential equations, and earlier work has also established RB approximations and associated rigorous a posteriori error bounds for the Stokes and Navier-Stokes equations when dealing with non-geometric parameters. Since the methodology readily admits certain classes of geometric variations, the theory has been extended to parametrized domains; however, in these earlier examples, either rigorous error bounds were not treated or only very simple geometric variations were considered.
The goal of the project if to further extend the RB method to deal with more generally parametrized incompressible fluid flow problems. Although such problems present additional difficulties in both theory and implementation, addressing these problems permits the use of the RB method for a wider, more relevant class of applications.
A-L Gerner, K Veroy, Reduced basis a posteriori error bounds for symmetric parametrized saddle point problems, submitted, preprint here.
A-L Gerner, K Veroy, Reduced basis a posteriori error bounds for the instationary Stokes equations, submitted, preprint here.
A-L Gerner, K Veroy, Certified reduced basis methods for parametrized saddle point problems, SIAM J. Sci. Comput., accepted, final preprint here.
A-L Gerner, K Veroy, Reduced basis a posteriori error bounds for the Stokes equations in parametrized domains: A penalty approach, Math. Models Methods Appl. Sci., 21 (2011), pp. 2103-2134.doi:10.1142/S0218202511005672 | final preprint here.
A-L Gerner, K Veroy, Reduced basis a posteriori error bounds for the instationary Stokes equations: A penalty approach, MATHMOD 2012 Conference Proceedings, accepted, final preprint here.