Preconditioners for regularized saddle point problems with an application for heterogeneous Darcy flow problems

Saddle point problems arise in the modelling of many important practical problems. Preconditioners for the corresponding matrices on block triangular form, based on coupled inner-outer iteration methods are analyzed and applied to a Darcy flow problem, possibly with strong heterogeneity and to non-symmetric saddle point problems. Using proper regularized forms of the given matrix and its preconditioner it is shown that, for large values of the regularization parameters, the eigenvalues cluster about one or two points on the real axis and that eigenvalue bounds do not depend on this variation. Therefore, just two outer iterations can suffice. To solve the inner iteration systems various preconditioners are used.