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Reduced Basis Methods (RBMs) are frequently proposed to approximateparametric problem solutions. They can be used to calculate solutions for alarge number of parameter values (e.g. for parameter fitting) as well as toapproximate a solution for a new parameter value (e.g. real time approximationwith a very high accuracy). They intend to reduce the computational costs ofHigh 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 codeexclusively as a "black-box," as opposed to other so-called intrusive methodsthat require code modification. This is very convenient when the HF code is acommercial one that has been purchased, as is frequently the case in theindustry. The effectiveness of this method relies on its decomposition into twostages, 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 thecontrary, the specificity of this NIRB approach is that, during the onlinepart, it solves the parametric problem on a coarse mesh only and then improvesits precision. As a result, it is significantly less expensive than a HFevaluation. This method has been originally developed for elliptic equationswith finite elements and has since been extended to finite volume. In thispaper, we extend the NIRB two-grid method to parabolic equations. We recoveroptimal 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 andon the Brusselator problem.

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