Augmented multiscale formulation for incompressible flow

We revise the multiscale finite element formulation for incompressible flow computations. We incorporate an iterative penalty approach to the classical multiscale finite element formulation to ensure the enforcement of the incompressiblity condition during temporal solution of the Navier–Stokes equations. The constraint of incompressibility, enforced by the continuity equation, is approached via an iterative penalty method. The multiscale formulation (MS) of Hughes et al. [1] is revised, and we introduce an Augmented Multiscale Formulation (AMS). We maintain consistency of the formulation by carrying velocity and pressure variables rather than an L² projection of the same as investigated by researchers earlier. Inexact Newton solvers are utilized to solve the resulting discrete problems. In this aspect, the formulation departs from the classical multiscale formulation for incompressible flow. In the present framework, it is not necessary to employ least-squares stabilization as an additional stabilization mechanism. This is consistent with the physics of incompressible flows, where shock waves and other strong singularities are absent. We validate the AMS formulation with Kovasznay flow problem, with benchmark solutions from literature. We benchmark the temporal evolutions of AMS formulation with the driven cavity solutions provided by the Augmented SUPG (ASUPG) and Augmented Galerkin least squares (AGLS) formulations. Further, we compare results obtained from ASUPG/AGLS formulations with the AMS formulation. The revised formulation is tested on a series of problems to serve as incompressible flow benchmarks and excellent agreement is reported for all problems with available experimental data. We report non-linear convergence histories of the formulation, as well which are difficult to provide within the MS formulation. The present AMS and MS formulations yield lower L² errors when compared with the analytical solutions of the Navier–Stokes equations. In a high order spectral/hp framework one has control over the polynomial order utilized. For low polynomial orders ( pₗₑᵥₑₗₛ of 3 and 4), the errors are also smaller than those obtained with the ASUPG formulation of Ranjan et al. [2]. For higher polynomial orders, however, the pressure exhibits suboptimal convergence.