Dual natural-norm a posteriori error estimators for reduced basis approximations to parameterized linear equations

Abstract

In this work, the concept of dual natural-norm for parametrized linear equations is used to derive residual-based a posteriori error bounds characterized by a O (1) stability constant. We translate these error bounds into very effective practical a posteriori error estimators for reduced basis approximations and show how they can be efficiently computed following an offline/online strategy. We prove that our practical dual natural-norm error estimator out-performs the classical inf-sup based error estimators in the self-adjoint case. Our findings are illustrated on anisotropic Helmholtz equations showing resonant behavior. Numerical results suggest that the proposed error estimator is able to successfully catch the correct order of magnitude of the reduced basis approximation error, thus outperforming the classical inf-sup based error estimator even for non self-adjoint problems.