Overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity

Low-order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill-conditioning, when the material approaches the incompressible limit, if only the displacement variables are used. Mixed finite elements using both displacement and pressure variables provide a well-known remedy, but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging. Two-level overlapping Schwarz preconditioners for the almost incompressible system of linear elasticity, discretized by mixed finite elements with discontinuous pressures, are constructed and analyzed. The preconditioned systems are accelerated either by a GMRES (generalized minimum residual) method applied to the resulting discrete saddle point problem or by a PCG (preconditioned conjugate gradient) method applied to a positive definite, although extremely ill-conditioned, reformulation of the problem obtained by eliminating all pressure variables on the element level. A novel theoretical analysis of the algorithm for the positive definite reformulation is given by extending some earlier results by Dohrmann and Widlund. The main result of the paper is a bound on the condition number of the algorithm which is cubic in the relative overlap and grows logarithmically with the number of elements across individual subdomains but is otherwise independent of the number of subdomains, their diameters and mesh sizes, the incompressibility of the material, and possible discontinuities of the material parameters across the subdomain interfaces. Numerical results in the plane confirm the theory and also indicate that an analogous result should hold for the saddle point formulation, as well as for spectral element discretizations.