Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

TL;DR

This paper extends the Non-Intrusive Reduced Basis two-grid method to parabolic equations, providing error estimates and demonstrating effectiveness through numerical results on heat and Brusselator problems.

Contribution

It introduces an extension of the NIRB two-grid method to parabolic equations and establishes optimal error estimates for this new application.

Findings

Achieved optimal error estimates in $L^{ abla}(0,T;H^1( abla))$ for parabolic equations.

Demonstrated the method's efficiency on heat and Brusselator problems.

Validated the approach with numerical experiments showing reduced computational costs.

Abstract

Reduced Basis Methods (RBMs) are frequently proposed to approximate parametric problem solutions. They can be used to calculate solutions for a large number of parameter values (e.g. for parameter fitting) as well as to approximate a solution for a new parameter value (e.g. real time approximation with a very high accuracy). They intend to reduce the computational costs of High Fidelity (HF) codes. We will focus on the Non-Intrusive Reduced Basis (NIRB) two-grid method. Its main advantage is that it uses the HF code exclusively as a "black-box," as opposed to other so-called intrusive methods that require code modification. This is very convenient when the HF code is a commercial one that has been purchased, as is frequently the case in the industry. The effectiveness of this method relies on its decomposition into two stages, one offline (classical in most RBMs as presented above) and one online. The offline part is time-consuming but it is only performed once. On the contrary, the specificity of this NIRB approach is that, during the online part, it solves the parametric problem on a coarse mesh only and then improves its precision. As a result, it is significantly less expensive than a HF evaluation. This method has been originally developed for elliptic equations with finite elements and has since been extended to finite volume. In this paper, we extend the NIRB two-grid method to parabolic equations. We recover optimal estimates in $L^{\infty}(0,T;H^1(\Omega))$ using as a model problem, the heat equation. Then, we present numerical results on the heat equation and on the Brusselator problem.