Abstract :
In this paper we investigate some domain decomposition methods for the solution of advection-diffusion (AD) equations. We aim at reviewing some well known methods and we introduce and analyze adaptive domain decomposition algorithms for nonoverlapping multidomain partitions. Two of the methods we will deal with were introduced by Cartenzoli and Quarteroni (1993) (Adaptive Dirichlet/Neumann (ADN) and Adaptive Robin/Neumann (ARN)) and are based on an iterative procedure among subdomains in which the transmission conditions take into account the direction of the streamlines. We introduce, also, a variant of ARN, d-ARN (damped-ARN). From the numerical point of view, these methods are applied using a finite element discretization. In particular, ARN and d-ARN perform very well, compared with the classical Dirichlet/Neumann method, when the equation is dominated by convection. We also present some studies related to the use of consistent stabilizations of the Galerkin method in a domain decomposition framework, and show how to enforce the transmission conditions in the presence of cross points.
