# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 1 Introduction  
The Navier-Stokes (NS) equations stand as the cornerstone of fluid dynamics, providing a mathematical description for the motion of viscous, incompressible fluids [1]. These equations, derived from the fundamental principles of conservation of mass and momentum, have profound implications across a vast spectrum of scientific and engineering disciplines. Their applications range from aerospace engineering, in the design of aircraft and spacecraft [2], and weather forecasting, where they model atmospheric and oceanic currents [3], to biofluid mechanics, for understanding blood flow in the cardiovascular system [4], and even in the creative industries for realistic visual effects [5]. Despite their ubiquity and fundamental importance, the inherent nonlinearity of the convective term,coupled with the pressure-velocity coupling, makes obtaining analytical solutions possible only in a few highly simplified cases [1,3]. Consequently, the advancement of science and technology has become increasingly reliant on numerical approximations to the NS equations, a field known as computational fluid dynamics (CFD) [2]. Among the various numerical discretization techniques, the finite element method (FEM) has emerged as a powerful and versatile tool for CFD [4,6]. Its strength lies in its ability to handle complex geometries and its solid mathematical foundation, rooted in the Galerkin method [7]. The standard Galerkin FEM seeks an approximate solution within a finite-dimensional function space by enforcing the residual of the governing equation to be orthogonal to the test function space [6]. For Stokes flow or low Reynolds number flows, this approach, when paired with appropriate velocity and pressure approximation spaces like the Taylor-Hood element (\(P_{2}-P_{1}\)) [8], which satisfies the crucial inf-sup (or Ladyzhenskaya-Babuska-Brezzi) condition, provides stable and accurate results [9,10]. However, the landscape changes dramatically when the flow Reynolds number (Re) becomes high as is common in many practical applications such as turbulence and boundary layer separation. In such regimes, the nonlinear convection term dominates the viscous diffusion term, leading to two major, intertwined challenges for numerical simulations. The first is a stability issue: the standard Galerkin method becomes inadequate for convection-dominated problems, often producing spurious, non-physical oscillations that can corrupt the entire solution domain [11,12]. To counteract this, stabilized finite element methods were developed. A pioneering and highly influential approach is the Streamline Upwind Petrov-Galerkin (SUPG) method [13], which introduces an additional term to the weighting function, acting along the streamline to add artificial diffusion only where needed, thus suppressing oscillations without overly compromising accuracy [14]. Building upon these ideas, the Variational Multiscale Stabilization (VMS) method offers a more sophisticated framework [15,16]. While SUPG effectively addresses oscillations by adding artificial diffusion along streamlines, VMS offers a more physically consistent approach by explicitly modeling the effect of unresolved subgrid scales on the resolved scales. This fundamental difference often leads to superior accuracy and less intrusive stabilization, particularly in complex, high-Reynolds-number flows where the interaction between scales is significant. Furthermore, compared to traditional Galerkin methods that often suffer from non-physical oscillations in convection-dominated regimes, the VMS component of the framework ensures the physical fidelity of the solution without excessive artificial diffusion. The core concept of VMS is the decomposition of the solution into large (resolved) scales and small (unresolved) subgrid scales. The effect of the unresolved scales on the resolved scales is then modeled, effectively damping numerical oscillations generated by high Reynolds numbers by explicitly constructing and accounting for subgrid-scale effects [17].