A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations
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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.
DepartmentMassachusetts Institute of Technology. Department of Mechanical Engineering
SIAM Journal on Scientific Computing
Society for Industrial and Applied Mathematics
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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