The condition number of the Schur complement in domain decomposition

In the nonoverlapping approach to domain decomposition (cf. [10], [21] and the references therein) the nite element equation for an elliptic boundary value problem is reduced, after parallel subdomain solves, to its Schur complement which involves only the nodal variables (degrees of freedom) on the subdomain boundaries. The condition number of the Schur complement has been investigated for second order elliptic problems and conforming nite elements in [3] and [18], and for a nite di erence scheme for the biharmonic equation on a rectangle with two subdomains in [11]. We present in this paper a general study of the condition number of the Schur complement in the context of nite element methods which covers elliptic problems of any order and both conforming and nonconforming nite elements. Let be a bounded open polyhedral domain IR (n = 2 or 3) and a( ; ) (the variational form of the elliptic boundary value problem) be a symmetric bilinear form on H( ) (m is a positive integer) which is de ned by an integral over . We assume that there exist positive constants C1 and C2 such that