A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations

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.