# A doubly reduced approximation for the solution to PDEs based on a domain truncation and a reduced basis method: Application to Navier-Stokes equations

Elise Grosjean,
Yvon Maday

February 2022
### Abstract

Reduced Basis Methods (RBM) are often proposed to approximate the solutions of a parameter-dependent problem for a large number of parameter values, as an alternative to classical solvers, in order to reduce the computational costs. They usually are decomposed in two stages. One stage is done offline and can be considered as a learning procedure where a Reduced Basis (RB) is built from several solutions, called snapshots, computed with a high fidelity (HF) classical method, involving, e.g. a fine mesh (finite element or finite volume). In the second stage, which is online and has to be very cheap, a reduced basis problem is solved. The efficiency of the RBM relies on the ability, offline, to prepare the online step. In this article, we consider a Non-Intrusive RBM (NIRB) which is the two grids method. One fine mesh is used to construct the snapshots for the generation of the RB. Then, in the online part, the NIRB algorithm involves a coarse mesh where the problem solution for a new parameter is approximated and is $L^2$-projected on the RB. We adapt the NIRB algorithm to improve the accuracy and to further reduce the complexity of the online part, in particular by using a \emph{domain truncation}. We exploit the fact that the solutions of parameterized problems behave physically similarly for a suitable range of parameters. We create two RB and a deterministic (algebraic) process which allows us to pass from one to the other. This new algorithm is applied to the 2D backward facing step. The second aim of this article is to deal with \emph{singularities}. The domain geometry produces a fluid recirculation zone that must be captured correctly, for instance with a refinement of the mesh. The two-grid method in the FEM context is applied with a new stategy in order to counterbalance the effects of the singularities. The channel domain considered in the model problem has one re-entrant corner and thus the convergence is not optimal with uniform meshes. Thus, the theory of the two-grid method does not applied. However, we present several numerical results with fine refined meshes where both NIRB approaches succeed in retrieving the fine FEM accuracy.

###### postdoctoral researcher

My research interests include numerics, P.D.E analysis, Reduced basis methods.