An improved error bound for reduced basis approximation of linear parabolic problems

Abstract

Abstract. We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β δ , the inverse of which enters into error estimates: β δ is unity for the heat equation; β δ decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.

A space-time hp-interpolation-based certified reduced basis method for Burgers' equation

“…The notation used in this section closely follows that of Urban and Patera. 12 We denote the triangulations of the temporal interval and spatial domain by T consists of N +1 elements with max κ∈T h diam(κ) ≤ h, belonging to a quasi-uniform family of meshes. Let us introduce a temporal trial space S ∆t , a temporal test space Q ∆t , and a spatial approximation space V h defined by…”

“…13,12 The space Y δ is equipped with the same inner product and the norm as the space Y. Our discrete approximation to Burgers' equation, Eq.…”

“…In particular, limited applicability of the classical a posteriori error bounding technique to unsteady Burgers' and Boussinesq equations are documented by Nguyen et al 9 and Knezevic et al 7 , respectively. In order to overcome the instability of the classical L 2 -in-time error-bound formulation, we follow the space-time approach recently devised by Urban and Patera, 13,12 ; we consider a space-time variational and corresponding finite element formulation that produces a favorable inf-sup stability constant and then incorporate the space-time truth within a space-time reduced basis approach. The approach is inspired by the recent work on the space-time Petrov-Galerkin formulation by Schwab and Stevenson 11 .…”

“…a The X -norm used in this work is slightly weaker than that used in Urban and Patera 13,12 that includes the terminal condition, w(T ) 2 H .…”

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…In [48] an improved h-p adaptive certified method is introduced to address the same natural convection problem, which has also been applied to a multiscale Stokes Fokker-Planck system modelling liquid crystals in [71]. More recent contributions in the field adopt a space-time Petrov-Galerkin variational approach to improve the control of the exponentially growing energy estimates in the linear case [123] dealing with convection-conduction problems, for Burgers' equations [131], Boussinesq equations for moderate Grashof number flows exhibiting steady periodic responses [129] and even addressing interesting hydrodynamic stability problems for moderate Reynolds number flows in an eddy-promoted channel [130].…”