I want to solve the following large system of linear equations: \begin{align} \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{c} x_{1} \\ x_{2} \end{array} \right]= \left[\begin{array}{c} b_{1} \\ b_{2} \end{array} \right] . \end{align}
Do people ever solve this system by iterating equations $ A_{11}x_{1} = b_{1} - A_{12} x_{2} $ and $ A_{22}x_{2} = b_{2} - A_{21} x_{1} $? It seems like this could be efficient, since we can calculate the LU decomposition of $ A_{11} $ and $ A_{22} $ faster than we can calculate the LU decomposition of $ \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] $.