Reduced Basis Methods and a Posteriori Error Estimators for Heat Transfer Problems

This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector — material properties, boundary conditions and sources, and geometry — we would like to compute certain outputs of engineering interest — heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The ∗Corresponding author (G. Rozza is research affiliate at MIT). §We thank Dr. D.B. Phuong Huynh (Singapore-MIT Alliance) and Dr. Paul Fisher (Argonne National Lab.) for their collaboration and we acknowledge Dr. Karen VeroyGrepl and Dr. Martin Grepl (formerly at MIT) for their previous contributions on natural convection. We thank the MIT Mechanical Engineering Pappalardo Book Fund for support of the educational components of this project. The development of the computational methodology is supported by AFOSR Grant No FA9550-051-0114 and FA-9550-07-1-0425. feasibility and efficiency of the method is demonstrated for three natural convection model problems: a scalar steady forced convection problem in a rectangular channel is characterized by two parameters — Péclet number and the aspect ratio of the channel — and an output – the average temperature over the domain; a steady natural convection problem in a laterally heated cavity is characterized by three parameters — Grashof and Prandtl numbers, and the aspect ratio of the cavity — and an output — the inverse of the Nusselt number; and an unsteady natural convection problem in a laterally heated cavity is characterized by two parameters — Grashof and Prandtl numbers— and a timedependent output — the average of the horizontal velocity over a specified area of the cavity.