Abstract:

This work focuses on the a posteriori error estimation for the reduced basis method
applied to partial differential equations with quadratic nonlinearity and affine
parameter dependence. We rely on natural norms (local parameter-dependent norms)
to provide a sharp and computable lower bound of the inf-sup constant.
We prove a formulation of the Brezzi-Rappaz-Raviart
existence and uniqueness theorem in the presence of
two distinct norms. This allows us to relax the existence condition and
to sharpen the field variable error bound.
We also provide a robust algorithm to compute the Sobolev embedding constants
involved in the error bound and in the inf-sup lower bound computation.
We apply our method to a steady natural convection problem in a closed cavity,
with Grashof number varying from 10 to 10⁷.