Two-scale Dirichlet–Neumann preconditioners for elastic problems with boundary refinements

The present work deals with the efficient resolution of elastostatics problems on domains with boundary refinements. The proposed approach separates the boundary refinements from the interior of the domain by the mortar method, and uses Dirichlet–Neumann preconditioners to solve the corresponding algebraic system. We prove that the simplest Dirichlet–Neumann algorithm achieves independence of the condition number of the preconditioned system with respect to the number and the size of the small details. Nevertheless, the situation no longer prevails when the refined boundary is clamped. An enhanced preconditioner is then designed by the introduction of a coarse space to mitigate the aforementioned sensitivity. Some numerical tests are performed to confirm the analysis, and the tools are extended by the proposition of a quasi-Newton method in the case of nonlinear elasticity. This paper is an extended version of a work presented at the DD16 conference with proofs and complete numerical results.