Colloque - Élise Grosjean : A Doubly Reduced Approximation for the Solution to PDEs Based on a Domain Truncation and a Reduced Basis Method: Application to Navier-Stokes Equations

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Élise Grosjean : A Doubly Reduced Approximation for the Solution to PDEs Based on a Domain Truncation and a Reduced Basis Method: Application to Navier-Stokes Equations

Élise Grosjean

Enseignante-chercheuse Inria, Équipe IDEFIX de l'Unité de Mathématiques Appliquées, ENSTA, Institut Polytechnique de Paris

Résumé

During this talk, I will present the NIRB two-grid method, together with recent extensions applied to the Navier–Stokes equations, aimed at further reducing the computational cost of the algorithm. The NIRB two-grid method, introduced in [1], is based on two stages. First, during an offline phase, a reduced basis is constructed from high-fidelity solutions computed on a fine mesh, involving a large number of degrees of freedom, using a standard discretisation technique. Then, during the online phase, the parametric problem is solved on a coarser mesh, and the resulting solution is projected onto the reduced space, thereby substantially decreasing the computational cost.

We extend this framework by further reducing the complexity of the online stage. As a representative application, we consider a classical benchmark problem in fluid mechanics: the two-dimensional Backward-Facing Step (BFS). In particular, we simplify the online computation by (i) using a coarse uniform mesh, rather than refining it near the re-entrant corner, and (ii) significantly truncating the outflow section of the channel. Both choices would typically be regarded as detrimental to the accuracy of a high-fidelity flow representation. To overcome this difficulty, we construct two reduced bases and introduce a deterministic linear mapping that enables the transfer from one basis to the other. Additional numerical simulations, including three-dimensional and time-dependent configurations, demonstrate the efficiency of the proposed approach.