Numerically stable LDL T - factorization of F - type saddle point matrices

We present a new algorithm that constructs a fill-reducing ordering for a special class of saddle point matrices: the F -matrices. This class contains the matrix occurring after discretization of the Stokes equation on a C-grid. The commonly used approach is to construct a fill-reducing ordering for the whole matrix and then change the ordering such that it becomes feasible. We propose to compute firstly a fillreducing ordering for an extension of the definite submatrix. This ordering can be easily extended to an ordering for the whole matrix. Herewith the construction of the ordering is straight forward and it can be computed efficiently with symbolic factoring. We show that a lot of structure of the matrix is preserved during Gaussian elimination. The preserved structure allows us to prove that any feasible ordering for an F -matrix is numerically stable. The growth factor is bounded by a number that depends linearly on the number of indefinite nodes. The algorithm allows for generalization to saddle point problems that are not of F -type and nonsymmetric, e.g. the incompressible Navier Stokes equations (with Coriolis force) on a C-grid. Numerical results for F -matrices show that the algorithm is able to produce a factorization with low fill.