A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations
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Yano-2014-A space-time Petrov.pdf
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Author(s)
Yano, Masayuki
Date Issued
February 2014
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Yano, Masayuki. “A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations.” SIAM Journal on Scientific Computing 36, no. 1 (February 20, 2014): A232–A266. © 2014, Society for Industrial and Applied Mathematics
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Final published version
Abstract
We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov--Galerkin space-time trial- and test-space norms that yield optimal and asymptotically mesh independent stability constants. We then employ an $hp$ Petrov--Galerkin (or minimum residual) space-time reduced basis approximation. We provide the Brezzi--Rappaz--Raviart a posteriori error bounds which admit efficient offline-online computational procedures for the three key ingredients: the dual norm of the residual, an inf-sup lower bound, and the Sobolev embedding constant. The latter are based, respectively, on a more round-off resistant residual norm evaluation procedure, a variant of the successive constraint method, and a time-marching implementation of a fixed-point iteration of the embedding constant for the discontinuous Galerkin norm. Finally, we apply the method to a natural convection problem governed by the Boussinesq equations. The result indicates that the space-time formulation enables rapid and certified characterization of moderate-Grashof-number flows exhibiting steady periodic responses. However, the space-time reduced basis convergence is slow, and the Brezzi--Rappaz--Raviart threshold condition is rather restrictive, such that offline effort will be acceptable only for very few parameters.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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DOI of Published Version
https://doi.org/10.1137/120903300
