A dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities utilizing linear local problems

Abstract

We propose a novel dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities. The proposed method is based on a particular Fenchel–Rockafellar dual formulation of the target problem, which yields linear local problems despite the nonlinearity of the target problem. Since local problems are linear, each iteration of the proposed method can be done very efficiently compared with usual nonlinear domain decomposition methods. We prove that the proposed method is linearly convergent with the rate 1−r−1/2 while the convergence rate of relevant existing methods is 1−r−1, where r is proportional to the condition number of the dual operator. The spectrum of the dual operator is analyzed for the cases of two representative variational inequalities in structural mechanics: the obstacle problem and the contact problem. Numerical experiments are conducted in order to support our theoretical results.

Additive Schwarz Methods for Convex Optimization as Gradient Methods

“…Such an idea of acceleration can be applied to not only linear problems but also nonlinear problems. There have been some recent works on the acceleration of domain decomposition methods for several kinds of nonlinear problems: nonlinear elliptic problems [12], variational inequalities [17], and mathematical imaging problems [15,16,19]. In particular, in the author's previous work [22], an accelerated additive Schwarz method that can be applied to the general convex optimization (1.1) was considered.…”