Balancing Neumann-Neumann Methods for Mixed Approximations of Linear Elasticity

Balancing Neumann-Neumann methods are introduced and analyzed for the algebraic systems of linear equations for mixed finite element approximations of linear elasticity for incompressible and almost incompressible materials as well as composite materials with different Lamé parameters in different parts of the domain. These methods solve iteratively the saddle point Schur complement, resulting from the implicit elimination of the interior degrees of freedom, using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. The resulting algorithm is very efficient, parallel, and robust with respect to material heterogeneities.