## 6 Numerical Methods and Algorithms
### Models at the Macro-Scale
#### 6.1.2 Solution of the \"Discrete in Space\"
The two main drawbacks are that \(A\) must be nonsingular and that the Schur complement \(S\) can be full, making it prohibitively expensive to factor or compute. When creating \(S\), numerical instability could potentially be an issue, particularly if \(A\) is not well conditioned [155]. If \(S\) is too costly to form or factor, a Schur complement reduction can still be used by solving related linear systems using iterative techniques like KSM, which only require \(S\) in the form of matrix-vector products that involve the matrices \(B_{1}^{T}\), \(B_{2}\), and \(C\), as well as by solving a linear system with matrix \(A\). These techniques do not require access to individual entries of \(S\). It is possible that the Schur complement system is not well conditioned, in which case preconditioning will be necessary. **Coupled methods** Coupled methods deal with the system (143) as a whole, computing \(x_{1}\) and \(x_{2}\) (or approximations to them) simultaneously. Among these approaches are direct solvers based on triangular factorizations of the global matrix \(\mathcal{A}\), as well as iterative algorithms such as KSM applied to the system as a whole (143), usually preconditioned in some way. For direct methods based on triangular factorizations of \(\mathcal{A}\), we give a brief overview limited to the symmetric case (that is, where \(A=A^{T}\), \(B_{1}=B_{2}=B\) and \(C=C^{T}\)), since no specialized direct solver exists for a nonsymmetric Saddle Point problem. Furthermore, we assume that \(A\) is positive definite and \(B\) has full rank, then the Saddle Point matrix \(\mathcal{A}\) admits the following factorization,  
\[\mathcal{A}=\mathcal{LDL}^{T} \tag{147}\]  
where \(\mathcal{D}\) is a diagonal matrix and \(\mathcal{L}\) is a unit lower triangular matrix. To be more precise, \(A\) is positive definite, so its decomposition is \(A=L_{A}D_{A}L_{A}^{T}\), where \(L_{A}\) is the unit lower triangular and \(D_{A}\) is the diagonal (also positive definite); additionally, the Schur complement \(\mathcal{S}=-(C+BA^{-1}B^{T})\) is negative definite, so its decomposition is \(S=-L_{S}D_{S}L_{S}^{T}\). Thus, we are able to write  
\[\mathcal{A}=\left[ \begin{array}{cc}A&B^{T}\\ B&-C\end{array} \right]=\left[ \begin{array}{cc}L_{A}&0\\ L_{B}&L_{S}\end{array} \right]\left[ \begin{array}{cc}D_{A}&0\\ 0&-D_{S}\end{array} \right]\left[ \begin{array}{cc}L_{A}^{T}&L_{B}^{T}\\ 0&L_{S}^{T}\end{array} \right]=\mathcal{LDL}^{T}, \tag{148}\]  
where \(L_{B}=BL_{A}^{-T}D_{A}^{-1}\). Note that \(BA^{-1}B^{T}=L_{B}D_{A}L_{B}^{T}\). However, in practice, with the original ordering, the factors will be fairly dense, and sparsity preservation requires the employment of symmetric permutations \(\mathcal{Q}\). Not every sparsity-preserving permutation is suitable, though. It is demonstrated that there are permutation matrices \(\mathcal{Q}\) such that the factorization of \(\mathcal{LDL}^{T}\), where \(\mathcal{D}\) is a diagonal matrix, is not available for \(\mathcal{QQ}^{T}\). Moreover, some permutations might result in issues with numerical instability [155]. Iterative methods are preferred for solving the linear system (143) using coupled methods, as sparse \(\mathcal{QQ}^{T}\) factorization methods are not completely foolproof, especially for Saddle Point systems arising from PDE problems. About the usage of iterative algorithms, we wrote some words about the use of preconditioned KSM for iterative solution of the entire linear system (143). Various preconditioning strategies designed for (generalized) Saddle Point systems are explained. To develop high-quality preconditioners for Saddle Point problems, one must take advantage of the problem's block structure and possess a comprehensive understanding of the origin and structure of each block [155]. We consider block diagonal and block triangular preconditioners for KSM applied to the coupled system \(\mathcal{A}x=y\) as in (143). The basic block diagonal preconditioner is given by  
\[\mathcal{P}_{d}=\left[ \begin{array}{cc}\hat{A}&0\\ 0&-\hat{S}\end{array}