Domain Decomposition Methods for Auxiliary Linear Problems of an Elliptic Variational Inequality

Abstract

Elliptic variational inequalities with multiple bodies 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. We describe two domain decomposition methods for solving such linear problems, namely, the FETI-FETI (finite element tearing and interconnecting) and hybrid methods, which are combinations of already existing domain decomposition methods. Estimates of the condition numbers of both methods are provided. The FETI-FETI method has a condition number which depends linearly on the number of subdomains across each body and polylogarithmically on the number of element across each subdomain. The hybrid method is a scalable alternative to the FETI-FETI method, and has a condition number with two polylogarithmic factors depending on the number of elements across each subdomain and across each body. We present numerical results confirming these theoretical findings.