The reduced basis method for incompressible viscous flow calculations
SIAM Journal on Scientific and Statistical Computing
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A reduced-order method for simulation and control of fluid flows
Journal of Computational Physics
SIAM Journal on Scientific Computing
Journal of Scientific Computing
"Natural norm" a posteriori error estimators for reduced basis approximations
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
SIAM Journal on Numerical Analysis
Reduced basis methods for Stokes equations in domains with non-affine parameter dependence
Computing and Visualization in Science
AMESOS: a set of general interfaces to sparse direct solver libraries
PARA'06 Proceedings of the 8th international conference on Applied parallel computing: state of the art in scientific computing
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
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This work focuses on the approximation of parametric steady Navier-Stokes equations by the reduced basis method. For a particular instance of the parameters under consideration, we are able to solve the underlying partial differential equations, compute an output, and give sharp error bounds. The computations are split into an offline part, where the values of the parameters are not yet identified, but only given within a range of interest, and an online part, where the problem is solved for an instance of the parameters. The offline part is expensive and is used to build a reduced basis and prepare all the ingredients - mainly matrix-vector and scalar products, but also eigenvalue computations - necessary for the online part, which is fast. We provide a model problem - describing natural convection phenomena in a laterally heated cavity - characterized by three parameters: Grashof and Prandtl numbers and the aspect ratio of the cavity. We show the feasibility and efficiency of the a posteriori error estimation by the natural norm approach considering several test cases by varying two different parameters. The gain in terms of CPU time with respect to a parallel finite element approximation is of three magnitude orders with an acceptable - indeed less than 0.1% - error on the selected outputs.