Structure-preserving Variational Multiscale Stabilization of the Incompressible Navier-Stokes Equations

This paper introduces a Variational Multiscale Stabilization (VMS) formulation of the incompressible Navier--Stokes equations that utilizes the Finite Element Exterior Calculus (FEEC) framework. The FEEC framework preserves the geometric and topological structure of continuous spaces and PDEs in the discrete spaces and model, and helps build stable and convergent discretizations. For the Navier-Stokes equations, this structure is encoded in the de Rham complex. In this work, we consider the vorticity-velocity-pressure formulation discretized within the FEEC framework. We model the effect of the unresolved scales on the finite-dimensional solution by introducing appropriate fine-scale governing equations, which we also discretize using the FEEC approach. This preserves the structure of the continuous problem in both the coarse- and fine-scale solutions; for instance, both the coarse- and fine-scale velocities are pointwise incompressible. We demonstrate that the resulting formulation is residual-based, energetically stable, and optimally convergent. Moreover, our fine-scale model provides an efficient computational approach: by decoupling fine-scale problems across elements, they can be solved in parallel. In fact, the fine-scale equations can be eliminated during matrix assembly, leading to a VMS formulation in which the problem size is governed solely by the coarse-scale discretization. Finally, the proposed formulation applies to both the lowest regularity discretizations of the de Rham complex and high-regularity isogeometric discretizations. We validate our theoretical results through numerical experiments, simulating both steady-, unsteady-, viscous-, and inviscid-flow problems. These tests show that the stabilized solutions are qualitatively better than the unstabilized ones, converge at optimal rates, and, as the mesh is refined, the stabilization is asymptotically turned off.