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 ${L}^{\mathrm{\infty}}(0,T;{H}^{1}(\mathrm{\Omega}))$ using as a model problem,
the heat equation. Then, we present numerical results on the heat equation and
on the Brusselator problem.

PREPRINT

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

Elise Grosjean and Yvon Maday

Submitted on 16 November 2022

## Abstract

## Preprint

Subject: Mathematics - Numerical Analysis