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.