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

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 \beta _{\delta } , the inverse of which enters into error estimates: β δ \beta _{\delta } is unity for the heat equation; β δ \beta _{\delta } 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.