I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form:

\begin{equation} A = \left[\begin{array}{c|c} \tilde{Q}&B^T \\ \hline B&*\\ \end{array}\right] = \left[\begin{array}{ccccc|c} Q & *& *& *&* \\ *& Q &*&*&* \\ *&*& \ddots&*&*& B^T\\ *&*&*& Q& * \\ *&*&*& *&Q \\ \hline & &B& & &*\\ \end{array}\right] , \end{equation} where $*$ means all zeros.

Let's say that $Q$ is a symmetric positive-definite $n$ by $n$ matrix and $B$ is $n*k$ by $m$ and full column rank. Both are **sparse** (small number of non-zeros per row).

I can factorize $A$ using sparse LDL decomposition but this doesn't take advantage of the repeated nature of the upper left corner.

Knowing that a factorization of $\tilde{Q}$ will just be a repeated block diagonal made up of factorizations of $Q$,
I tried using the Schur complement method, but then it seems I need to factorize $B \tilde{Q}^{-1} B^T$ which could be **dense**.

Is there a way to build a LU-style factorization for $A$ (for efficient solving) which takes advantage of both the sparsity of $Q$ and $B$ and the repeated block diagonal in $\tilde{Q}$?