Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Abstract. Balancing Neumann-Neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coeÆcients, discretized with mixed nite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point Schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. A polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coeÆcient case. Parallel and serial numerical experiments con rm the theoretical results, indicate that they still hold for systems with discontinuous coeÆcients, and show that our algorithm is scalable, parallel, and robust with respect to material heterogeneities. The results on heterogeneous general problems are also supported in part by our theory.