Abstract: Problems on saddle point form arise in several applications when solving partial differential equations.They are frequently ill-conditioned. Conditions for non-singularity of the corresponding finite element matrices are presented and it is shown how to regularize the problem if they are singular or nearly singular. Efficient preconditioners on block tridiagonal form are presented. It is shown that they give a strong clustering of the eigenvalues of the preconditioned matrix about just one or two points on the real axis.This holds also for non-symmetric problems. Spectral equivalence and compact perturbation properties are shown for the corresponding differential operator pairs. These results show a mesh independent and superlinear rate of convergence of the iterative solution method. Inner iteration methods for the regularized matrices are discussed and some numerical illustrations of the methods for heterogeneous material coefficient problems show the practical bearing of the methods.
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