# Time discretization of the variational formulation of the Navier-Stokes equation

I've asked this question on mathoverflow too.

Let

• $T>0$
• $I:=(0,T]$
• $d\in\mathbb N$
• $\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{H^1(\Lambda,\:\mathbb R^d)}}\;,\;\;\;H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{L^2(\Lambda,\:\mathbb R^d)}}$$
• $\operatorname P_H$ denote the orthogonal projection from $L^2(\Lambda,\mathbb R^d)$ onto $H$
• $A_0u:=-\Delta u$ for $u\in\mathcal D(A_0):=H_0^1(\Lambda,\mathbb R^d)\cap H^2(\Lambda,\mathbb R^d)$, $$Au:=\operatorname P_HA_0u\;\;\;\text{for }u\in\mathcal D(A):=\mathcal D(A_0)\cap V$$ and $$B(u,v):=(u\cdot\nabla)v\;\;\;\text{for }u\in L^2(\Lambda,\mathbb R^d)\text{ and }v\in H^1(\Lambda,\mathbb R^d)$$
• $f:I\to H$
• $u\in L^2(I,\mathcal D(A))$ with $u'\in L^2(I,H)$ and $$u'(t)+A_0u(t)+B(u(t),u(t))+\nabla p(t)=f(t)\;\;\;\text{for all }t\in I\tag1$$ for some $p:I\to H^1(\Lambda)$

Assuming that $\Lambda$ is sufficiently regular such that $(1)$ is well-defined, it can be shown that $(1)$ is equivalent to $$u'(t)+Au(t)+\operatorname P_HB(u(t),u(t))=f(t)\;\;\;\text{for all }t\in I\;.\tag2$$ I want to solve $(2)$ numerically and I'm only interested in $u$ (and not in $p$).

I know that there are many references for the numerical study of $(1)$. However, it seems to me that all the considered schemes don't use $(2)$. They only use $(2)$ for theoretical results like existence and uniqueness of solutions. Maybe I'm wrong and I just don't see that these schemes use $(2)$.

In any case, my question is: Are we able to provide a numerical scheme which solves $(2)$ directly?

Or is there something which prevents us from doing that? My idea is to apply, for example, a semi-implicit Oseen discretization in time, i.e. consider $$\frac{u(t_n)-u(t_{n-1})}h+Au(t_n)+\operatorname P_HB(u(t_{n-1}),u(t_n))=f(t_n)\;\;\;\text{for all }n\in\left\{1,\ldots,N\right\}\tag3$$ with $$t_n:=nh\;\;\;\text{for }n\in\left\{0,\ldots,N\right\}$$ and $h:=T/N$ for some $N\in\mathbb N$. After that for each $n\in\left\{1,\ldots,N\right\}$ $(3)$ should be solvable by a finite element method (or is there some problem that I don't see?).

• A quick google search turned up these: epubs.siam.org/doi/abs/10.1137/030601533 sciencedirect.com/science/article/pii/0022247X85903300 I'm not sure if they're what you're looking for. – David Ketcheson Feb 25 '17 at 17:04
• @DavidKetcheson Unfortunately not. I'm looking for a reference where the time discretization of $(1)$ is tested against divergence-free test functions. The construction of a practicable scheme where the finite element functions are exactly divergence-free seems to be nontrivial. – 0xbadf00d Feb 25 '17 at 17:44
• Are you aware of Nedelec elements? Are they not what you're looking for? – David Ketcheson Feb 25 '17 at 19:55
• @DavidKetcheson No, I wasn't and, until now, I'm not sure if they are what I'm looing for. tThe motivation for my question was the following: In a dissertation (see the fourth paragraph at page 38 (in the PDF-ordering)), I've read that "it is nontrivial to construct practicable numerical schemes where finite el ement functions are exactly divergence-free". I just don't understand why this is nontrivial. – 0xbadf00d Feb 25 '17 at 23:48
• @DavidKetcheson I could imagine, that the reason is due to the approximation of the pressure (in which I'm not interested). After a time discretization has ben chosen, I don't see the problem to consider weak formulations as they are stated in this lecture note (see page 50 (in the PDF-ordering)). – 0xbadf00d Feb 25 '17 at 23:48

The something that prevents me from using such a scheme with projections is the numerical realization of the projector $P_H\colon L^2 \to H$.

• I don't know of a formulation that is better accessible to numerical algorithms than expression of the projection via $$v_0 = P_Hv \leftrightarrow \begin{bmatrix}I & \nabla \\ div & 0\end{bmatrix} \begin{bmatrix}v_0 \\ *\end{bmatrix} = \begin{bmatrix}v \\ 0\end{bmatrix},$$ where the $*$ is a dummy variable,

• which I would approximate by means of mixed finite elements,

• which (probably) would realize $(3)$ as a standard mixed FEM of the Navier-Stokes in the $(v,p)$-formulation.

So... If you have a reasonable numerical realization of $P_H$, there is nothing wrong with the scheme sketched in $(3)$.

• You don't need a realization of $\operatorname P_H$. If we test $(1)$ against $v\in V$, we obtain $$\frac1h\langle u^n-u^{n-1},v\rangle_H+\mathfrak a(u^n,v)+\mathfrak b(u^{n-1},u^n,v)=\langle f^n,v\rangle_H\;,$$ where $u^n:=u(t_n)$, $f^n:=f(t_n)$, $$\mathfrak a(u,v):=\sum_{i=1}^d\langle\nabla u_i,\nabla v_i\rangle_{L^2(\Lambda,\:ℝ^d)}\;\;\;\text{for }u,v\in H_0^1(Λ,ℝ^d)$$ and $$\mathfrak b(u,v,w):=\langle B(u,v),w\rangle_{L^2(Λ,\:ℝ^d)}\;\;\;\text{for }u,v,w\in H_0^1(Λ,ℝ^d)\;.\tag 4$$ – 0xbadf00d Feb 27 '17 at 10:45
• The crucial point is that testing $(3)$ against $v$ yields $(3)$ too. So, this simply corresponds to choosing a finite element space of divergence-free functions. (cf. chapter 5 of this lecture notes) – 0xbadf00d Feb 27 '17 at 10:46
• So... what is your question.... Do divergence-free test functions exist and why does no one use them?? We have added some discussion and references on this issue in our paper: Altmann/H. Finite element decomposition and minimal extension for flow equations (postprint on my homepage) – Jan Feb 28 '17 at 8:25
• I've asked a question on the steady Navier-Stokes problem in the hope that things are clearer over there. In the enumeration of that question, I wonder why everybody is using $(1)$ and hence is dealing with mixed finite elements, while we could also use $(2)$ and hence would only need to deal with the space $V$. – 0xbadf00d Feb 28 '17 at 12:38
• No, the introduction. – Jan Feb 28 '17 at 13:51