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

In this paper we further explore the non-intrusive reduced basis (NIRB) approach known as the two-grid method . This technique is designed to efficiently simulate parametric partial differential equations by significantly reducing the computational cost of high-fidelity models, especially in scenarios requiring solutions for a large number of parameter values or real-time computations. As with other reduced basis methods, the “offline stage” in which the reduced basis is constructed relies on a classical discretization technique (e.g., the finite element method) with a large number of degrees of freedom to ensure a sufficiently accurate high-fidelity approximation. What distinguishes the two-grid method and makes it non-intrusive is that, during the “online stage”, the same discretization method is employed, but with a significantly reduced number of degrees of freedom, thereby greatly lowering the computational cost. We here extend this idea by further reducing the complexity of the online stage. As an example of application, we consider a classical fluid problem, the 2D Backward Facing Step (BFS). We simplify the online step by i) using a coarse uniform mesh instead of refining it at the re-entrant corner and ii) significantly truncating the outflow part of the channel, two choices that would normally be considered in opposition to obtaining a high fidelity representation of the flow. To accomplish this, we create two reduced bases and a linear deterministic process that allows us to pass from one to the other. Additional numerical simulations (3D, time-dependent) illustrate the efficiency of this new approach.