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

For approximating parametric problem solutions, Reduced Basis Methods (RBMs) are frequently proposed. They intend to reduce the computational costs of High Fidelity (HF) codes while maintaining the HF accuracy. They generally require an offline/online decomposition and a significant modification of the HF code in order for the online computation to be performed in short (or even real) time. We focus on the Non-Intrusive Reduced Basis (NIRB) two-grid method in this paper. Its main feature is the use of the HF code as a “black-box” on a coarse mesh for a new parameter during the online process, followed by an accuracy improvement based on the reduced basis paradigm that is realized in a very short time. Unlike other more intrusive methods, this approach does not necessitate code modification. As a result, it costs significantly less than an HF evaluation. This method was developed for elliptic equations with finite elements and has since then been extended to finite volume schemes. In this paper, we extend the NIRB two-grid method to parabolic equations. We recover optimal estimates in natural norms, using the heat equation as a model problem and present several numerical results on the heat equation and on the Brusselator problem.