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There is a huge literature on efficient preconditioners for saddle-point problems. In computational physics, the case where the Lagrange multipliers enjoy a weak formulation (say, the Stokes equations) is almost always considered. In such cases, very fast solvers can be designed.

Unfortunately, in structural mechanics, the Lagrange multipliers are often introduced on the discrete level. Consider for instance multi-points constraints between degrees of freedom (this feature is very common in industrial software). I know this is not the good way of doing things, but this is how it is often done.

I know about the ancient work of St-Georges, Notay, Warzée, or the recent work of Rees and Greif, Gould, Hribar and Nocedal. But I wonder if someone has experienced other strategies (such as the ones available in the FieldSplit preconditioners of PETSc for instance) for preconditioning saddle point systems with those horrible discrete Lagrange multipliers? I am indded seeking a way of using efficient preconditioners (say multigrid) on these saddle-point systems.

Finally, I wonder if the best way to use efficient preconditioners with Lagrange multipliers wouldn't be the domain decomposition methods, such as FETI or BDD, that can naturally handle multipliers.

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