# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 3 Numerical Methods
### Variational Multiscale Stabilization
The specific powers of the mesh size \(h_{K}\) (i.e., \(h_{K}^{2}\) for the element residual and \(h_{K}\) for the jump term) arise from the scaling arguments in the a posteriori error analysis, ensuring that the estimator \(\eta_{K}\) correctly reflects the local contribution to the global error in the energy norm. (Adapted from the dual-weighted residual framework of [PERSON] and [PERSON] (2001) [30].) While the fundamental structure of the posteriori error estimator, which balances element residuals and inter-element flux jumps, adapts principles from well-established frameworks for finite element methods (e.g., the dual-weighted residual framework cited from [PERSON] and [PERSON] [30]), its novelty and key contribution in this paper lie in its specific formulation and application within the integrated VMS-AMR system. Specifically, the estimator \(\eta_{K}\) is not applied to the original Navier-Stokes equations. Instead, it is crucially formulated based on the residuals \(\mathbf{R}_{M}\) and \(\mathbf{R}_{\mathbb{C}}\) of the VMS-stabilized large-scale equations, which inherently include the subgrid-scale model terms (\(\mathbf{S}_{M}^{n+\theta}\) and \(\mathbf{S}_{\mathbb{C}}^{n+\theta}\)). This design choice is pivotal for the synergy of the framework. It creates a self-consistent feedback loop: the adaptive mesh refinement is directly guided by the quality of the stabilized solution itself. Consequently, the mesh is refined not only in regions with high physical gradients (e.g., boundary layers and vortices) but also in areas where the subgrid-scale model is most active or exhibits significant residuals. This ensures that computational resources are precisely allocated to improve the fidelity of the coupled VMS-AMR solution, a more nuanced and targeted approach than applying a standard estimator to an unstabilized formulation. With the posteriori error estimator (13) in hand, adaptive mesh refinement can been implemented. The following is a dynamic refinement strategy for adaptive meshes:  
1. Calculate \(\eta_{K}\) for all cells;
2. Set a threshold \(\lambda\in(0,1)\), and mark the units that satisfy \(\eta_{K}>\lambda\max\eta_{K}\);
3. Generate a new mesh using the longest-edge bisection method ([PERSON] (1984) [31]) or local remeshing ([PERSON] et al. (1983) [32]). (Longest-edge bisection: bisect the longest edge of the element, generating a new node. Local remeshing: generate a finer Delaunay triangulation in the marked region.) The overall workflow of the proposed integrated VMS-AMR framework is summarized in Figure 1 and Algorithm 1 below. This process illustrates the feedback loop where the solution from the VMS-stabilized system is used to calculate an error estimator, which in turn guides the adaptive mesh refinement for the next time step, ensuring that computational effort is dynamically focused on the most critical regions of the flow. ```
1:Initialize:
2:Set time \(t\gets 0\), time step counter \(n\gets 0\). 3:Define final time \(T\), time step size \(\Delta t\), and AMR threshold \(\lambda\in(0,1)\). 4:Generate initial mesh \(\mathcal{T}^{0}_{h}\). 5:Set initial velocity \(\mathbf{u}^{0}_{h}\) and pressure \(p^{0}_{h}\) on \(\mathcal{T}^{0}_{h}\). 6:while\(t<T\)do
7:\(n\gets n+1\), \(t\gets t+\Delta t\). 8: (1) Solve the VMS system for \((\mathbf{u}^{n}_{h},p^{n}_{h})\) on mesh \(\mathcal{T}^{n-1}_{h}\)
9: Solve the nonlinear system (e.g., using Picard or Newton iteration) based on Equation (9) to find the large-scale solution \((\mathbf{u}^{n}_{h},p^{n}_{h})\). 10: This involves computing stabilization parameters and subgrid-scale model terms. 11: (2) The Posteriori Error Estimation
12:for all element \(K\in\mathcal{T}^{n-1}_{h}\)do
```  
**Algorithm 1** Integrated VMS-AMR framework for unsteady Navier-Stokes equations. \[\left.\
u\Delta\mathbf{u}_{h}^{n})+\
abla p_{h}^{n}\right)- \mathbf{f}^{n+\theta}+\mathbf{S}_{M}^{n+\theta},\]4 A flowchart of the integrated VMS-AMR framework. 13.