I've asked this question on Math.SE too.Let$d\in\left\{1,\ldots,4\right\}$$\Lambda\subseteq\mathbb R^d$ be bounded, nonempty and open and $\partial\Lambda$ be Lipschitz$V:=\left\{u\in H_0^1(\Lambda,\mathbb R^d):\nabla\cdot u=0\right\}$$W:=\left\{p\in L^2(\Lambda):\int_\Lambda p=0\right\}$$\mathfrak a(u,v):=\sum_{i=1}^d\langle\nabla u_i,\nabla v_i\rangle_{L^2}$ for $u,v\in H^1(\Lambda,\mathbb R^d)$$\mathfrak b(p,v):=\langle p,\nabla\cdot v\rangle_{L^2}$ for $p\in L^2(\Lambda)$ and $v\in H^1(\Lambda,\mathbb R^d)$$\mathfrak c(u,v,w):=\langle((u\cdot\nabla)v,w\rangle_{L^2}$ for $u,v,w\in H^1(\Lambda,\mathbb R^d)$The usually studied variational formulation of the steady Navier-Stokes equation is\begin{equation}\left\{\begin{split}\mathfrak a(u,v)+\mathfrak b(p,v)+\mathfrak c(u,u,v)&=0\;\;\;\text{for all }v\in H_0^1(\Lambda,\mathbb R^d)\\\mathfrak b(u,q)&=0\;\;\;\text{for all }q\in W\end{split}\tag1\right.\end{equation}where $(u,p)\in H_0^1(\Lambda,\mathbb R^d)\times W$ is the searched solution.I want to solve $(1)$ numerically. Usually, $(1)$ is linearized in some way and then a mixed finite element method is used to approximate a solution.However, I'm not interested in $p$. Now, $$\mathfrak a(u,v)+\mathfrak c(u,u,v)=0\;\;\;\text{for all }v\in V\tag2$$ (where $u\in V$ is the searched solution) is a variational formulation of the steady Navier-Stokes equation which is equivalent to $(1)$ and which doesn't contain $p$.I wonder if it might be better for me to find a numeric scheme which solves $(2)$. A linearization, e.g. an Oseen iteration, is possible for $(2)$ too. I think the crucial point is the choice of a (conforming) finite element.So, the question if I can benefit from the fact that I'm not interested in $p$ and hence don't need to care about its approximation. I've read many papers, but I couldn't find any which tries to solve the steady Navier-Stokes equation for the velocity only.
Numerical computation of the velocity in the steady Navier-Stokes equation
finite element;pde;numerical analysis;fluid dynamics;variational calculus