# Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
## 4 Block Preconditioners for the Navier-Stokes Discretized Systems
### Block Triangular Preconditioner  
We will now analyze the Block Triangular Preconditioner (BTP) applied to the discretized Navier-Stokes equations. Let us define \(H\) the \(n+m\times n+m\) Navier-Stokes coefficient matrix and \(M\) the preconditioner:  
\[H=\begin{bmatrix}F&B^{T}\\ B&0\end{bmatrix},\qquad\text{and}\quad M=\begin{bmatrix}M_{F}&B^{T}\\ 0&-M_{S}\end{bmatrix},\]  
with \(M_{F}\) and \(M_{S}\) scalable preconditioners for \(F\) and for the negative Schur complement \(S=BF^{-1}B^{T}\), respectively. From now on, scalability will denote the property of a given preconditioner to produce roughly the same number of iterations as the mesh is refined. In other words, the eigenvalues of the preconditioned matrices will be uniformly bounded in \(h\).  
To analyze the eigenvalues of the preconditioned matrix \(HM^{-1}\mathbf{w}=\lambda\mathbf{w}\), we assume that there exist LU factorizations of \(M_{F}=L_{F}U_{F}\) and \(M_{S}=L_{S}U_{S}\) and define  
\[\Pi_{L}=\text{diag}(L_{F},L_{S}),\qquad\text{and}\quad\Pi_{U}=\text{diag}(U_{ F},U_{S}).\]  
Subsequently, the eigenvalue problem is equivalent to  
\[\Pi_{L}^{-1}H\Pi_{U}^{-1}\mathbf{z}=\lambda\Pi_{L}^{-1}M\Pi_{U}^{-1}\mathbf{z},\qquad \mathbf{z}=\Pi_{U}\mathbf{w}.\]  
Defining \(N=L_{F}^{-1}B^{T}U_{S}^{-1},R=L_{S}^{-1}BU_{F}^{-1}\) and exploiting the blocks, we obtain  
\[\begin{bmatrix}F_{P}&N\\ R&0\end{bmatrix}\begin{bmatrix}\mathbf{z}_{1}\\ \mathbf{z}_{2}\end{bmatrix}=\lambda\begin{bmatrix}I_{n}&N\\ 0&-I_{m}\end{bmatrix}\begin{bmatrix}\mathbf{z}_{1}\\ \mathbf{z}_{2}\end{bmatrix} \tag{11}\]where \(F_{P}=L_{F}^{-1}FU_{F}^{-1}\) is similar to the preconditioned \((1,1)\) block \(M_{F}^{-1}F\).  
Matrices \(R\) and \(N\) satisfies:  
\[RN=L_{S}^{-1}BU_{F}^{-1}L_{F}^{-1}B^{T}U_{S}^{-1}=L_{S}^{-1}BF_{P}^{-1}B^{T}U_{S }^{-1}=L_{S}^{-1}\hat{S}U_{S}^{-1}\equiv S_{P},\]  
where \(\hat{S}=B^{T}M_{F}^{-1}B\) is the **exact** (minus) Schur complement related to the matrix  
\[\begin{bmatrix}M_{F}&B^{T}\\ B&0\end{bmatrix},\]  
and hence \(S_{P}\) represents this Schur complement, preconditioned by the spectrally equivalent matrix \(M_{S}\). Observing that the matrix on the right in (11) is involutory we rewrite (11) as  
\[\begin{bmatrix}F_{P}&N\\ R&0\end{bmatrix}\begin{bmatrix}I_{n}&N\\ 0&-I_{m}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix}=\lambda\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix},\qquad\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix}=\begin{bmatrix}I_{n}&N\\ 0&-I_{m}\end{bmatrix}\begin{bmatrix}\mathbf{z}_{1}\\ \mathbf{z}_{2}\end{bmatrix}, \tag{12}\]  
which finally reads  
\[H_{BTP}\mathbf{u}=\lambda\mathbf{u}\qquad\text{or}\quad\begin{bmatrix}F_{P}&(F_{P}-I_{ n})N\\ R&RN\end{bmatrix}\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix}=\lambda\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix}. \tag{13}\]  
Writing now \(E_{F}=F_{P}-I_{n}\) and \(E_{S}=S_{P}-I_{m}\), (13) rewrites as  
\[\begin{pmatrix}\begin{bmatrix}I_{n}&0\\ R&I_{m}\end{bmatrix}+\begin{bmatrix}E_{F}&0\\ 0&E_{S}\end{bmatrix}\begin{bmatrix}I_{n}&N\\ 0&I_{m}\end{bmatrix}\end{pmatrix}\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix}=\lambda\begin{bmatrix}\mathbf{u}_{1}\\ \mathbf{u}_{2}\end{bmatrix}. \tag{14}\]  
We have shown that the BTP-preconditioned matrix is similar to a matrix that can be written as one with all eigenvalues at one plus another one whose norm is bounded by the norms of \(E_{F}\) and \(E_{S}\).