Reorthogonalization-based stiffness preconditioning in FETI algorithms with applications to variational inequalities

Abstract

A cheap symmetric stiffness-based preconditioning of the Hessian of the dual problem arising from the application of the finite element tearing and interconnecting domain decomposition to the solution of variational inequalities with varying coefficients is proposed. The preconditioning preserves the structure of the inequality constraints and affects both the linear and nonlinear steps, so that it can improve the rate of convergence of the algorithms that exploit the conjugate gradient steps or the gradient projection steps. The bounds on the regular condition number of the Hessian of the preconditioned problem, which are independent of the coefficients, are given. The related stiffness scaling is also considered and analysed. The improvement is demonstrated by numerical experiments including the solution of a contact problem with variationally consistent discretization of the non-penetration conditions. The results are relevant also for linear problems.