Block-Triangular Preconditioners for Saddle Point Problems with a Penalty Term

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Abstract
Block-triangular preconditioners for a class of saddle point problems with a penalty term are considered. An important example is the mixed formulation of the pure displacement problem in linear elasticity. It is shown that the spectrum of the preconditioned system is contained in a real, positive interval and that the interval bounds can be made independent of the discretization and penalty parameters. This fact is used to construct bounds of the convergence rate of the GMRES method with respect to an energy norm. Numerical results are given for GMRES and BI-CGSTAB.