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

We present a space-time certified reduced basis method for long-time integration of parametrized noncoercive parabolic equations with quadratic nonlinearity. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov-Galerkin space-time trialand test-space norms which yields 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-Grashofnumber flows exhibiting steady periodic responses.