\begin{align}
\mymat{A}\myvec{u}=\myvec{f}, \qquad \myvec{u},\myvec{f} \in \mathbb{R}^{N}, \quad \mymat{A} \in \mathbb{R}^{N\times N}, \label{ch5:eq:poissonsystem}
\end{align}
-where $\mymat{A}$ is the sparse matrix formed by finite difference coefficients, $N$ is the number of unknowns, and $\myvec{f}$ is given by \eqref{ch5:eq:poissonrhs}. Equation \eqref{ch5:eq:poissonsystem} can be solved in numerous ways, but a few observations may help do it more efficiently. Direct solvers based on Gaussian elimination are accurate and use a finite number of operations for a constant problem size. However, the arithmetic complexity grows with the problem size by as much as $\mathcal{O}(N^3)$ and does not exploit the sparsity of $\mymat{A}$. Direct solvers are therefore mostly feasible for dense systems of limited sizes. Sparse direct solvers exist, but they are often difficult to parallelize, or applicable for only certain types of matrices. Regardless of the discretization technique, the discretization of an elliptic PDE into a linear system as in \eqref{ch5:eq:poissonsystem} yields a very sparse matrix $\mymat{A}$ when $N$ is large. Iterative methods\index{iterative methods} for solving large sparse linear systems find broad use in scientific applications, because they require only an implementation of the matrix-vector product, and they often use a limited amount of additional memory. Comprehensive introductions to iterative methods may be found in any of~\cite{ch5:Saad2003,ch5:Kelley1995,ch5:Barrett1994}.
+where $\mymat{A}$ is the sparse matrix formed by finite difference coefficients, $N$ is the number of unknowns, and $\myvec{f}$ is given by \eqref{ch5:eq:poissonrhs}. Equation \eqref{ch5:eq:poissonsystem} can be solved in numerous ways, but a few observations may help do it more efficiently. Direct solvers based on Gaussian elimination are accurate and use a finite number of operations for a constant problem size. However, the arithmetic complexity grows with the problem size by as much as $\mathcal{O}(N^3)$ and does not exploit the sparsity of $\mymat{A}$. Direct solvers are therefore mostly feasible for dense systems of limited sizes. Sparse direct solvers exist, but they are often difficult to parallelize, or applicable for only certain types of matrices. Regardless of the discretization technique, the discretization of an elliptic PDE into a linear system as in \eqref{ch5:eq:poissonsystem} yields a very sparse matrix $\mymat{A}$ when $N$ is large. Iterative methods\index{iterative method} for solving large sparse linear systems find broad use in scientific applications, because they require only an implementation of the matrix-vector product, and they often use a limited amount of additional memory. Comprehensive introductions to iterative methods may be found in any of~\cite{ch5:Saad2003,ch5:Kelley1995,ch5:Barrett1994}.
One benefit of the high abstraction level and the template-based library design is to allow developers to implement their own components, such as iterative methods for solving sparse linear systems. The library includes three popular iterative methods: conjugate gradient, defect correction\index{defect correction}, and geometric multigrid. The conjugate gradient method is applicable only to systems with symmetric positive definite matrices. This is true for the two-dimensional Poisson problem, when it is discretized with a five-point finite difference stencil, because then there will be no off-centered approximations near the boundary. For high-order approximations ($\alpha>1$), we use the defect correction method with multigrid preconditioning. See e.g., \cite{ch5:Trottenberg2001} for details on multigrid methods.
