Convergence Analysis of a Domain Decomposition Method with Augmented Lagrangian Formulation

Abstract

Through the combination of an augmented Lagrangian formulation with a preconditioned inexact Uzawa algorithm, we construct a domain decomposition based method for finite element approximation of linear second-order elliptic partial differential equations. With this approach, the proposed method shares the main features of Lagrange multipliers based domain decomposition methods, i.e. number of iterations bounded by the local element size (H/h) using a simple coarse space and direct application to decompositions with irregular sub domain geometry, with the advantage that inexact solvers at sub domain level are allowed at sub domain level. An analysis of the method applied to the Poisson equation is presented to justify the preconditioners choices and to derive bounds for the convergence factor in function of the local element size.