Domain decomposition preconditioners for multiscale problems in linear elasticity

We analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al.1 The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor.