When there are two variables in a PDE to be solved, the system matrix for the discretized variational formulation or weak form is a \(2\times 2\) block matrix instead of a single block,

\[\begin{equation} \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} u \\ w \end{pmatrix} = \begin{pmatrix} f_1 \\ f_2 \end{pmatrix}. \end{equation}\]

Such a system matrix appears in the boundary integral equation for the Laplace problem with mixed boundary condition:

\[\begin{equation} \begin{pmatrix} \gamma_0^{\rm int} u \\ \gamma_1^{\rm int} u \end{pmatrix} = \begin{pmatrix} (1-\sigma) I - K & V \\ D & \sigma I + K' \end{pmatrix} \begin{pmatrix} \gamma_0^{\rm int} u \\ \gamma_1^{\rm int} u \end{pmatrix} + \begin{pmatrix} N_0 f \\ N_1 f \end{pmatrix}. \end{equation}\]

Schur complement is often used to solve this equation system.

  1. Multiply the equation in first row with \(-M_{21}M_{11}^{-1}\) and add the result to the second row.

    \[\begin{equation} \begin{pmatrix} M_{11} & M_{12} \\ 0 & M_{22} - M_{21}M_{11}^{-1}M_{12} \end{pmatrix} \begin{pmatrix} u \\ w \end{pmatrix} = \begin{pmatrix} f_1 \\ f_2 - M_{21}M_{11}^{-1}f_1 \end{pmatrix}. \end{equation}\]

    Here, \(M_{22} - M_{21}M_{11}^{-1}M_{12}\) is called the Schur complement of \(M\) with respect to the block \(M_{11}\).

  2. Solve the following equation for \(w\).

    \[\begin{equation} \left( M_{22} - M_{21}M_{11}^{-1}M_{12} \right) w = f_2 - M_{21}M_{11}^{-1} f_1. \end{equation}\]

  3. Solve the following equation for \(u\).

    \[\begin{equation} \label{eq:2} M_{11} u = f_1 - M_{12}w. \end{equation}\]

We can see that the Schur complement method is actually the Gauss-Seidel (Gauss elimination) method for a \(2\times 2\) matrix, or the method for solving linear equations of two variables which we once learnt in middle school. Sometimes a high level mathematical method, such as the Schur complement used in PDE, tallies with an elementary method, both of which bear a same idea or spirit.

The benefit of this method is by decomposing the original large system matrix into two smaller problems, less memory is needed and the iterative solver usually converges faster.