Two Domain Decomposition Methods for Auxiliary Linear Problems of a Multibody Elliptic Variational Inequality

Abstract

Elliptic variational inequalities with multiple bodies in two dimensions are considered. It is assumed that an active set method is used to handle the nonlinearity of the inequality constraint, which results in auxiliary linear problems. For solving such linear problems we study two domain decomposition methods called the finite element tearing and interconnecting (FETI-FETI) and hybrid methods in this paper. Bodies are decomposed into several subdomains in both methods. The FETI-FETI method combines the one-level FETI and the dual-primal FETI (FETI-DP) methods. We present a proof that this combined method has a condition number that depends linearly on the number of subdomains across each body and polylogarithmically on the number of elements across each subdomain. Our numerical results, and those of others, suggest that this is the best possible bound. The hybrid method combines the one-level FETI and the balanced domain decomposition by constraints (BDDC) methods; we prove that the condition number of this method has two polylogarithmic factors depending on the number of elements across each subdomain and across each body. We present numerical results confirming this theoretical finding.Key words. domain decomposition, variational inequalities, one-level finite element tearing and interconnecting, dual-primal finite element tearing and interconnecting, balanced domain decomposition by constraints 1. Introduction. We consider domain decomposition methods for elliptic variational inequalities with multiple bodies. It is assumed that an active set method, such as the primal-dual active set strategy in [24], is used to reduce the variational inequality into a sequence of linear problems; we focus on solving such auxiliary linear problems in a scalable manner and examine two domain decomposition methods for this purpose. Both methods combine existing domain decomposition methods, and each body is decomposed into subdomains that in turn are unions of finite elements in both methods.The first method, called the finite element tearing and interconnecting (FETI-FETI) method in this paper, is a combination of the original FETI (one-level FETI) and the dual-primal FETI (FETI-DP) methods. We prove that the condition number of the FETI-FETI method depends linearly on the number of subdomains across each body and polylogarithmically on the number of elements across each subdomain under some restrictions. Currently we have this result only in two dimensions (2D); 1 see section 3.1 for details. Such results were previously shown numerically in [29] and [30] without a proof.The second method, called the hybrid method in this paper, is a combination of the FETI-DP and the balanced domain decomposition by constraints (BDDC)

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

“…There is a sizable literature on domain decomposition methods for second order variational inequalities [25,38,5,33,35,36,34,4,27,12,3,26]. (References for related multigrid methods can be found in the survey article [20].)…”

On conditioning of Schur complements of H-TFETI clusters for 2D problems governed by Laplacian

“…The details of implementation are a bit tricky, but well known, see e.g. Klawonn and Rheinbach [10], Lee [11], or Dostál et al [4], Chap. 19.…”

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

“…In designing domain decomposition methods for either (1) or (2), one might assume that the nonlinear problem is reduced to a sequence of linear problems by, for example, the active set strategy 25 . Then domain decomposition can be applied to each linear problem in that sequence 26 . However, we do not adopt such a strategy since it does not affect the convergence rate of outer iterations.…”

“…25 Then domain decomposition can be applied to each linear problem in that sequence. 26 However, we do not adopt such a strategy since it does not affect the convergence rate of outer iterations. Instead, we are interested in nonlinear domain decomposition; domain decomposition is applied directly to the nonlinear problem (1).…”