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Pré-Publication, Document De Travail Année : 2023

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

Résumé

This paper focuses on the non-intrusive reduced basis (NIRB) method called \textit{the two grids method}. It is used for the simulation of parametric Partial Differential Equations (PDEs) to reduce the associated computational costs of a High-Fidelity (HF) code when such problems must be solved for a large number of parameter values. As other Reduced Basis Methods (RBM), its effectiveness is based on an offline/online decomposition of its algorithm. The offline part is time-consuming but only done once. Then, in the online stage, a considerably less expensive problem is solved. The term "non-intrusive" refers to the fact that its algorithm does not necessitate any change of the HF code, as it is the case with so-called intrusive RBM. Its particularity comes from the use of two meshes: one coarser mesh is employed during the online portion of the algorithm.\\ We modify the NIRB two-grid algorithm to improve its accuracy and reduce the runtimes of its online part, particularly by employing a \emph{domain truncation}. To accomplish this, we create two RB and a deterministic process that allows us to pass from one to the other. This new algorithm is applied to a classical fluid problem, the 2D Backward Facing Step (BFS), and several numerical results are presented.\\ Then, we apply a new strategy to efficiently deal with \emph{domain singularities}. Indeed, because the channel domain in the BFS problem has one re-entrant corner, convergence with uniform meshes is not optimal. The theory of the two-grid method is no longer applicable, but this new approach allows us to counterbalance the effects of singularities while avoiding the cost of a mesh refinement during the online stage. We provide several numerical results to illustrate its potential for time reduction.
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Dates et versions

hal-03588508 , version 1 (24-02-2022)
hal-03588508 , version 2 (26-01-2023)
hal-03588508 , version 3 (22-03-2023)

Identifiants

  • HAL Id : hal-03588508 , version 2

Citer

Elise Grosjean, Yvon Maday. A doubly reduced approximation for the solution to PDE's based on a domain truncation and a reduced basis method: Application to Navier-Stokes equations. 2023. ⟨hal-03588508v2⟩
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