# Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
## 2 Problem Setting and Discretization
### Weak Formulation  
The general formulation of Equations (1) and (2), which require as solution a function \(\mathbf{u}\) twice differentiable and a function \(p\) continuously differentiable, can be relaxed, allowing for solutions that satisfy weaker requirements. Sometimes, in fact, there does not exist any solution that satisfies the strong form of the equations, while it is possible to find weak solutions.  
Before deriving the weak form of the Navier-Stokes equations, let us define the space of square-integrable functions  
\[L^{2}(\Omega)=\{f\colon\Omega\mapsto\mathbb{R}\ |\ \int_{\Omega}|f(\mathbf{x})|^{2}\,d \Omega<+\infty\}\]  
and the Sobolev Space  
\[H^{k}(\Omega)=\{f\in L^{2}(\Omega)\ |\ D^{\bullet}f\in L^{2}(\Omega)\quad\forall \alpha\colon|\alpha|\leq k\},\]  
i.e., the space of square integrable functions whose derivatives up to order \(k\) are still square integrable. Here, \(D^{\bullet}f\) represents a weak derivative.  
Now, starting from the stationary Navier-Stokes equations, it is possible to obtain the weak formulation multiplying by a test function \(\mathbf{v}\in V\), where the space \(V\) will be definesd later, and integrating over \(\Omega\).  
\[-\int_{\Omega}\
u\Delta\mathbf{u}\cdot\mathbf{v}\,d\Omega+\int_{\Omega}[( \mathbf{u}\cdot\
abla)\mathbf{u}]\cdot\mathbf{v}\,d\Omega+\int_{\Omega} \
abla p\cdot\mathbf{v}\,d\Omega=\int_{\Omega}\mathbf{f}\cdot\mathbf{v}\,d \Omega\quad\forall\mathbf{v}\in V.\]  
In this form, there is still the necessity for the function \(\mathbf{u}\) to be twice differentiable, which is the condition that we want to relax. In order to do so, we exploit [PERSON]'s formulas (i.e., integration by parts in multiple dimensions) and rewrite the integral involving \(\Delta\mathbf{u}\) as the sum of an integral involving only \(\
abla\mathbf{u}\) and a boundary integral. The same can be done for the pressure term. The result is  
\[\int_{\Omega}\
u\
abla\mathbf{u}\cdot\
abla\mathbf{v}\,d\Omega+ \int_{\Omega}[(\mathbf{u}\cdot\
abla)\mathbf{u}]\cdot\mathbf{v}\,d\Omega-\int _{\Omega}p\,\
abla\cdot\mathbf{v}\,d\Omega=\\ \int_{\Omega}\mathbf{f}\cdot\mathbf{v}\,d\Omega+\int_{\partial \Omega}\Bigl{(}\
u\frac{\partial\mathbf{u}}{\partial\mathbf{n}}-p\mathbf{n} \Bigr{)}\cdot\mathbf{v}\,dS\quad\forall\mathbf{v}\in V. \tag{3}\]  
The same can be done for the second equation, when considering a test function \(q\in Q\). The result is  
\[\int_{\Omega}q\,\
abla\cdot\mathbf{u}\,d\Omega=0\quad\forall q\in Q. \tag{4}\]  
Accordingly, an alternative version of the Navier-Stokes equations is: find \(\mathbf{u}\in V\) and \(p\in Q\) such that (3) and (4), together with the proper initial and boundary conditions, hold for every possible choice of \(\mathbf{v}\in V\) and \(q\in Q\). The couple \((\mathbf{u},p)\) is then called a weak solution of the Navier-Stokes equations. For this formulation to make sense, all the integrals involved must be well defined: this is achieved by choosing properly the spaces \(V\) and \(Q\). The correct choice is to set \(V\) equal to the space of functions in \(H^{1}(\Omega)\) such that they satisfy the Dirichlet boundary condition on the Dirichlet portion of the boundary; Q should simply be the space \(L^{2}(\Omega)\). It can be checked that, in this way, all of the integrals are well defined (see [15]).