# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 1 Introduction
The second major challenge is computational efficiency. High-Reynolds-number flows are inherently multiscale, featuring a wide range of spatial and temporal scales, from large energy-containing eddies down to the smallest dissipative structures at the Kolmogorov scale [18]. Resolving all these scales with a uniformly fine computational grid, a technique known as Direct Numerical Simulation (DNS), is prohibitively expensive and often computationally intractable for practical engineering problems [19]. This necessitates a more intelligent approach to mesh generation. Adaptive mesh refinement (AMR) provides a compelling solution to this problem [20]. Guided by a posteriori error estimators, which measure the local error of the numerical solution, AMR techniques dynamically adjust the mesh resolution, placing smaller elements in regions of high flow complexity (e.g., steep gradients, vortices, and boundary layers) and coarser elements elsewhere [21,22]. This strategy significantly reduces the total number of degrees of freedom and, consequently, the computational cost, while maintaining a high level of accuracy where it is most needed [23]. While both VMS and AMR are powerful techniques in their own right, the existing research has often applied them independently, thereby failing to fully exploit their potential synergistic advantages. For instance, many studies have successfully applied VMS to simulate turbulent flows, but without adaptive meshing, they often require globally fine grids, leading to a significant increase in computational effort that contributes little to the accuracy of the results [24]. Conversely, adaptive methods that lack a proper stabilization treatment, such as the one described by [PERSON] (2013) [23], can easily lead to numerical oscillations in convection-dominated scenarios, even on an adapted mesh. The true potential for a breakthrough in efficiency and accuracy lies in their combination: using AMR to focus computational power on the dynamically important regions of the flow, while employing VMS to ensure the stability and correctness of the solution on the resulting anisotropic and non-uniform grids. While previous works have explored aspects of VMS or AMR independently, or even combined them in a less integrated fashion, the distinct novelty of this paper lies in the deep and self-consistent coupling of these two powerful techniques. Unlike traditional approaches where error estimators might operate on the original Navier-Stokes residuals (as in some standard AMR frameworks), this framework introduces a novel posteriori error estimator specifically formulated based on the dual residuals of the VMS-stabilized large-scale equations. This ensures that the adaptive mesh refinement is directly guided by the quality of the VMS solution itself, including the effects of the subgrid-scale model. This integrated design, where stabilization and adaptation mechanisms work in concert rather than as separate, sequential steps, represents a significant advancement over existing methodologies. The innovation of this paper lies in the construction of such an enhanced Galerkin framework that deeply couples VMS with an AMR strategy. This synergy is realized through a novel posteriori error estimator specifically designed for the VMS system. Unlike traditional estimators that might act on the original Navier-Stokes residuals, the estimator (as detailed in Section 3.3) is formulated based on the dual residuals of the stabilized large-scale equations. This ensures that the adaptive mesh refinement is directly guided by the quality of the VMS solution itself, including the effects of the subgrid-scale model. This approach creates a truly integrated and self-consistent framework where the stabilization and adaptation mechanisms work in concert, rather than as separate, sequential steps. Based on the standard Ritz-Galerkin finite element method and the stable Taylor-Hood element, a VMS term is introduced, derived from a generalized \(\theta\)-scheme for temporal discretization.