Isogeometric Schwarz preconditioners for linear elasticity systems

Isogeometric Schwarz preconditioners are constructed and analyzed for both compressible elasticity in primal formulation and almost incompressible elasticity in mixed formulation. These preconditioners require the solution of local elasticity problems on overlapping subdomains forming a decomposition of the problem domain and the solution of a coarse elasticity problem associated with the subdomain coarse mesh. An h-analysis of the preconditioner for the primal formulation of compressible elasticity yields an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes. Extensive numerical experiments in 2D and 3D confirm this theoretical bound and show that an analogous bound holds for the mixed formulation of almost incompressible elasticity. The numerical tests also show the good preconditioner performance with respect to the polynomial degree p and regularity k of the isogeometric basis functions, as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation.