Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

Abstract

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.

Appendix: Proof of the Inverse Inequalities (8)

To prove inequalities (8), we follow very closely the arguments developed in [8, Thm. 4.5.11].

We first need to introduce an equivalent norm for fractional order Hilbert spaces as follows. Let $s\in (0,1)$. Then

$$\begin{aligned} \Vert u\Vert ^2_{H^s(\Omega )}=\Vert u\Vert ^2+|u|^2_{H^s(\Omega )}, \end{aligned}$$

where

$$\begin{aligned} |u|^2_{H^s(\Omega )}=\int _\Omega \int _\Omega \frac{|u(\varvec{x})-u(\varvec{y})|^2}{|\varvec{x}-\varvec{y}|^{3+2s}}\,\mathrm{d}\varvec{x}\,\mathrm{d}\varvec{y}. \end{aligned}$$

Given $(K,{\mathcal {P}}, \Sigma )$, we define $(\tilde{K}, \tilde{P}, \tilde{\Sigma })$ where $\hat{K}=\{(1/h_K)\varvec{x}: \varvec{x}\in K\}$. Thus, if $u_h$ is a function defined on K, then $\hat{u}_h$ is defined on $\tilde{K}$ by

$$\begin{aligned} \hat{u}(\hat{\varvec{x}})=u ( h_K^{-1}\varvec{x})\quad {\text { for all }} \quad \hat{\varvec{x}}\in \hat{K}. \end{aligned}$$

Thus we can write

$$\begin{aligned} \Vert \nabla u_h\Vert _{\varvec{L}^2(K)}= h_K^{\frac{1}{2}} \Vert \hat{\nabla }\hat{u}_h\Vert _{\varvec{L}^2(K)}. \end{aligned}$$

As $\hat{\nabla }\hat{u}_h$ belongs to a space of finite and fixed dimension on $\hat{K}$, on which all norms are equivalent, it is not hard to see that there is a constant $C_{\hat{T}}>0$ such that

$$\begin{aligned} \Vert \hat{\nabla }\hat{u}_h\Vert _{\varvec{L}^2(K)}\le C_{\hat{T}} |\hat{u}_h|_{H^s(\hat{K})}. \end{aligned}$$

Reverting to K, this leads to

$$\begin{aligned} \Vert \hat{\nabla } \hat{u}\Vert _{\varvec{L}^2}\le C_{\hat{T}} h_K^{-\frac{3}{2}+s} |u_h|_{H^{s}(K)} \end{aligned}$$

and hence

$$\begin{aligned} \Vert \nabla u_h\Vert _{\varvec{L}^2(K)}\le C_{\hat{T}} h^{-1+s}_K \Vert u_h\Vert _{H^s(\Omega )}. \end{aligned}$$

An argument in the proof of [8, Prop. 4.4.11] shows that if $(\tilde{K}, \tilde{{\mathcal {P}}}, \tilde{\Sigma })$ is a referent element, we have that there exists a constant $C_{\tilde{T}}>0$ such that $C_{\hat{T}}\le C_{\tilde{T}}$. Summing over all elements K and using the quasi-uniformity of the mesh leads to

$$\begin{aligned} \Vert \nabla u_h\Vert \le C h^{-1+s} \left( \sum _{K\in {\mathcal {T}}_h} \Vert u_h\Vert _{H^s(\Omega )}^2\right) . \end{aligned}$$

Then (8) follows because the sum of the fractional norms over all elements is smaller than the fractional norm over the union of the elements.