# 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

I implemented a semi-implicit Oseen discretization in time numerical scheme using Mathematica FEM and tested it on the problems of thermal convection and flow around an aerodynamic profile. Coincidence with other methods is generally good. A detailed report along with the code can be viewed at https://community.wolfram.com/groups/-/m/t/1433064 I also used a similar numerical scheme for compressible and turbulent flows. For example, 3D turbulent flow in a rectangular channel with jump section https://mathematica.stackexchange.com/questions/217202/transition-to-turbulence

My opinion is that nothing wrong with this algorithm as I tested it and compare with other numerical methods.

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