Keywords: domain decomposition, FETI-DP, quadratic programming, scalable algorithms, contact problems.

Domain decomposition methods are powerful iterative methods for solving systems of algebraic equations arising from finite element discretizations of elliptic partial differential equations and those of the FETI type belong to the most successful. Our research concerns the development of the scalable FETI-based methods for contact problems. An algorithm is called scalable for a given class of problems if it has both, parallel and numerical scalabilities. Parallel scalability is characterized by the speed up due to parallel implementation nearly proportional to the number of processors and numerical scalability by the cost of solution proportional to the number of nodal parameters.

The FETI-DP is based on the decomposition into non-overlapping subdomains, where the continuity of the primal solution at crosspoints is implemented directly into the formulation of the primal problem so that one degree of freedom is considered at each crosspoint and the continuity of the solution on auxiliary interfaces is enforced by Lagrange multipliers. After eliminating the primal variables, the problem reduces to a small, relatively well conditioned strictly convex QP problem that is solved iteratively. For semi-coercive problems the efficiency of the FETI-DP can be further improved by introducing special projectors onto an auxiliary space related to rigid body modes of floating bodies and preconditioners: lumped and Dirichlet.

FETI methods are even more successful for the solution of variational inequalities. The reason is that duality transforms the general inequality into the nonnegativity constraints so that efficient algorithms that exploit inensive projections and other tools may be exploited. For nonlinear problems the scalability of FETI-DPC based on active set strategies with additional planning steps was established by Farhat et al. only experimentally. Dostál et al. proved this scalability theoretically.

Farhat et al. observed that the corner nodes on a contact interface cause difficulties and recommended avoiding them. The generalization of the method for corners on contact interfaces is described for variational inequalities through the additional condition that preserves the nonpenetration in Lagrange multipliers. This richer corner mesh results in better convergence of the method because of better error propagation across the nonlinear interface and in better preconditioning of the nonlinear steps using standard FETI-DP preconditioners. The numerical experiments confirm the scalability of the algorithm for contact problems. We showed that for unpreconditioned and preconditioned FETI-DP using no corners on contact zone the numbers of conjugate gradient (CG) iterations increase much faster with increasing number of subdomains along the contact interface in comparison to the case we use corners on the contact zone, when the numbers of CG iterations vary very moderately. The results demonstrate that for a given decomposition the use of corners always significantly reduces the number of CG iterations for both unpreconditioned and preconditioned systems and this effect is magnified with an increasing number of subdomains along the contact interface, i.e. with increasing number of corners on the the contact zone.

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