Reduced basis approximation and a posteriori error estimation for a Boltzmann model

We present a reduced basis approximation and associated rigorous a posteriori output (flow rate) error bound for a simple linearized BGK Boltzmann model for flow in a channel; we consider two parameters, the Knudsen number and the accommodation coefficient. The reduced basis approximation is built upon, and the reduced basis error is measured relative to, a “truth” (mapped, weighted) spectral discretization of a stabilized Petrov–Galerkin weak formulation. We exploit the affine nature of the parametric dependence to provide an offline–online computational procedure that decouples the spectral and reduced basis quantities: the computational complexity of the online stage – in which, given a new parameter value, we evaluate the reduced basis flow rate prediction and associated error bound – depends only on the (typically very small) dimension of the reduced basis space, N, and not   on the (typically very large) dimension of the “truth” spectral approximation, NN. We demonstrate that the reduced basis approximation converges very rapidly with N, and that the error estimator effectivities remain bounded albeit somewhat large. In the online stage, the reduced basis approximation achieves a certifiable relative accuracy of 10−2 with respect to the “truth” spectral result at only 1/100th the computational effort. The method is appropriate in the many-query and real-time (e.g., parameter estimation) contexts in which there is a premium on reduced marginal cost – and a tolerance for significant offline (pre-processing) effort.