True Error Control for the Localized Reduced Basis Method for Parabolic Problems
For researchers in numerical methods for PDEs, this work generalizes a previously existing method to a new class of problems (parabolic), representing an incremental advance.
The paper extends the localized reduced basis method to parabolic problems with true error certification, providing offline/online decomposable a posteriori error estimates that account for both discretization and model reduction errors.
We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques [10, 9, 3, 5]. In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method [14]. Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.