I have the following equation: $$ \left[ \begin{array}{ccc} K_{mm}&K_{mn}&-I\\ K_{nm}&K_{nn}&0\\ V_C&0&I_C\\ \end{array} \right]\left[ \begin{array}{c} x_m\\ x_n\\ i_m\\ \end{array} \right]=\left[ \begin{array}{c} 0\\ 0\\ b_c\\ \end{array} \right] $$
where $m$ is about several hundred, $n$ is about several million, $I_{m*m}$ is unit matrix and $V_C$, $I_C$ are square matrix with $m*m$, $K_{nn}$ is symmetric sparse matrix and we can get $$ K_{nn}=L'L$$ and $L$ is the lower triangular matrix by Cholesky decomposition, $$V_C=\left[ \begin{array}{ccc} 1&0&...&-1\\ 0&1&...&-1\\ ...&...&...&...\\ 0&...&1&-1\\ 0&0&...&0\\ \end{array} \right]$$ $$I_C=\left[ \begin{array}{ccc} 0&...&0\\ ...&...&...\\ 0&...&0\\ 1&...&1 \end{array} \right]$$
Is there any efficient method to solve this sparse matrix by $L$?
Thanks,
Tang Laoya