Domain decomposition methods for nonlinear problems in fluid dynamics

The numerical solution of finite element approximations of complicated two- and three-dimensional nonlinear problems can be a most formidable task. In order to overcome this difficulty related to dimensionality, domain-splitting methods can be very effective, particularly in view of obtaining a fast and economical conjugate gradient solver, which can be used to precondition the solution of nonlinear problems by optimization methods via nonlinear least squares or weighted residual formulations. A new technique of this type will be introduced and analyzed and its efficiency will be discussed from numerical experiments concerning the numerical simulation of transonic flows for compressible inviscid fluids and incompressible viscous flows modeled by the Navier-Stokes equations.