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Fictitious components and subdomain alteranting methods

The paper describes one of the versions of the fictitious components method for elliptic boundary value problems in the generalized formulation. The conditions are defined for the convergence of the method constructed, and the convergence rate estimates are given. The Schwartz subdomain alternating method which is dual with respect to the fictitious components method is also considered, and the conditions for its convergence are formulated. The paper ends with illustrating the convergence rate for the methods proposed in the one-dimensional case. The fictitious components method as an iterative process for solving systems of difference and variational difference equations was proposed and investigated in [1,3,4,7-9]. This method can be treated in a certain sense as a closure of the fictitious domain method [14]. This paper describes one of the versions of the fictitious components method whose formulations for solving elliptic-t>pe boundary value problems and their variational difference approximations are coincident [9,10]. The method dual to the fictitious components method is a version of the well-known subdomain alternating method [9-11]. An analysis of variational difference counterparts of these methods in subspaces [5] is of primary importance in their justification and realization. 1. FICTITIOUS COMPONENTS METHOD Let us illustrate the idea of construction of the fictitious components method. Let i2Q, Ωι and Ω2 be bounded domains in R, such that