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 settings requiring solutions for many parameter values or real-time computations. As in other reduced basis methods, the offline stage, during which the reduced basis is constructed, relies on a classical discretization technique, such as the finite element method, with a large number of degrees of freedom in order to obtain sufficiently accurate high-fidelity approximations. What distinguishes the two-grid method, and makes it non-intrusive, is that during the online stage the same discretization method is used on a much coarser mesh, thereby substantially reducing the computational cost. Here, we extend this idea by further reducing the complexity of the online stage. As a model application, we consider a classical fluid dynamics problem: the two-dimensional backward-facing step (BFS). We simplify the online step by (i) using a coarse uniform mesh instead of refining the mesh near the re-entrant corner and (ii) significantly truncating the outflow part of the channel. Both choices would normally be expected to compromise the high-fidelity representation of the flow. To overcome this difficulty, we construct two reduced bases and introduce a deterministic linear mapping that transfers information from one basis to the other. Additional numerical simulations, including three-dimensional and time-dependent cases, illustrate the efficiency of this new approach.