Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment

TL;DR

This paper develops a localized, adaptive error estimation and enrichment technique for the reduced basis multi-scale method applied to parametric elliptic problems, improving computational efficiency and solution accuracy.

Contribution

It introduces a localized, conservative flux-based error estimator with adaptive online enrichment for the LRBMS method, enhancing error control and efficiency.

Findings

Effective error bounds via flux reconstruction

Localized enrichment improves solution accuracy

Numerical experiments confirm method applicability

Abstract

In this contribution we consider localized, robust and efficient a-posteriori error estimation of the localized reduced basis multi-scale (LRBMS) method for parametric elliptic problems with possibly heterogeneous diffusion coefficient. The numerical treatment of such parametric multi-scale problems are characterized by a high computational complexity, arising from the multi-scale character of the underlying differential equation and the additional parameter dependence. The LRBMS method can be seen as a combination of numerical multi-scale methods and model reduction using reduced basis (RB) methods to efficiently reduce the computational complexity with respect to the multi-scale as well as the parametric aspect of the problem, simultaneously. In contrast to the classical residual based error estimators currently used in RB methods, we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solution. In addition, the resulting error estimator is localized and can thus be used in the on-line phase to adaptively enrich the solution space locally where needed. The resulting certified LRBMS method with adaptive on-line enrichment thus guarantees the quality of the reduced solution during the on-line phase, given any (possibly insufficient) reduced basis that was generated during the offline phase. Numerical experiments are given to demonstrate the applicability of the resulting algorithm with online enrichment to single phase flow in heterogeneous media.