# An Improved Galerkin Framework for Solving Unsteady High-Reynolds Navier-Stokes Equations
## 3 Numerical Methods
### Variational Multiscale Stabilization
Since \(\Pi_{h}\mathbf{u}\) is a polynomial of degree \(P_{k}\), the higher-order terms (\(|\alpha|=k+1\)) are truncated; hence  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]2  
Using \(|\mathbf{x}-\mathbf{x}_{c}|\leq h_{K}\) and \(|D^{k+1}\mathbf{u}|\leq|\mathbf{u}|_{H^{k+1}(K)}\), there is  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]3  
Similarly, the gradient error \(\
abla\mathbf{e}_{\mathbf{u}}^{I}\) is given by  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]4  
Take \(h=\max_{K}\{h_{K}\}\), and sum over all cells  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]5  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]6  
Therefore, for quasi-uniform meshes (\(h_{K}\approx h\)), the interpolation error estimate is obtained as  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]7  
where \(h_{K}\) is the element size. The pressure error \(\|\mathbf{e}_{p}\|_{L^{2}}\) is bounded by  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]8  
where \(\beta\) is a constant independent of the mesh size. Substituting (10) into the momentum equation (2) yields  
\[\mathbf{R}_{M}:= \left(\frac{\mathbf{u}_{h}^{n+1}-\mathbf{u}_{h}^{n}}{\Delta t}+ \theta((\mathbf{u}_{h}^{n+1}\cdot\
abla)\mathbf{u}_{h}^{n+1}-\
u\Delta \mathbf{u}_{h}^{n+1}+\
abla p_{h}^{n+1})+(1-\theta)((\mathbf{u}_{h}^{n}\cdot \
abla)\mathbf{u}_{h}^{n}-\right.\]9  
Therefore, it can be obtained that the pressure error \(\|\mathbf{e}_{p}\|_{L^{2}}\) is controlled by the velocity error \(\|\mathbf{e}_{\mathbf{u}}\|_{H^{1}}\) as  
\[\left.\
u\Delta\mathbf{u}_{h}^{n})+\
abla p_{h}^{n}\right)- \mathbf{f}^{n+\theta}+\mathbf{S}_{M}^{n+\theta},\]0By substituting the error expression (11) into the error estimation on each element \(K\), based on the estimates of the residual and boundary jump terms, as well as the interpolation error estimate (12), the local energy norm control can be obtained as  
\[\left.\
u\Delta\mathbf{u}_{h}^{n})+\
abla p_{h}^{n}\right)- \mathbf{f}^{n+\theta}+\mathbf{S}_{M}^{n+\theta},\]1  
Summing the local estimates over all cells, there is  
\[\left.\
u\Delta\mathbf{u}_{h}^{n})+\
abla p_{h}^{n}\right)- \mathbf{f}^{n+\theta}+\mathbf{S}_{M}^{n+\theta},\]2  
Using discrete stability conditions (such as the inf-sup condition), global posteriori error estimator can be ultimately obtained as  
\[\left.\
u\Delta\mathbf{u}_{h}^{n})+\
abla p_{h}^{n}\right)- \mathbf{f}^{n+\theta}+\mathbf{S}_{M}^{n+\theta},\]3  
where \(\eta_{K}\) is the local error indicator.