A Note on the LDLT Decomposition of Matrices from Saddle-Point Problems

Abstract

Sparse linear systems Kx=b are considered, where K is a specially structured symmetric indefinite matrix. These systems arise frequently, e.g., from mixed finite element discretizations of PDE problems. The LDLT factorization of K with diagonal D and unit lower triangular L is known to exist for natural ordering of K, but the resulting triangular factors can be rather dense. On the other hand, for a given permutation matrix P, the LDLT factorization of PTKP may not exist. In this paper a new way to obtain a fill-in minimizing permutation based on initial fill-in minimizing ordering is introduced. For an important subclass of matrices arising from mixed and hybrid finite element discretizations, the existence of the LDLT factorization of the permuted matrix is proved. Experimental results on practical problems indicate that the amount of computational savings can be substantial when compared with the approach based on Schur complement.