Preconditioned iterative methods for algebraic saddle-point problems

In this paper, we consider the preconditioned Lanczos method for the numerical solution of algebraic systems with singular saddle point matrices. These systems arise from algebraic systems with singularly perturbed symmetric positive definite matrices. The original systems are replaced by equivalent systems with saddle point matrices. Two approaches are proposed to design preconditioners for singular saddle point matrices. The algorithms are applied to the diffusion equation with strongly heterogeneous and anisotropic diffusion tensors.