Convergence of domain decomposition methods via semi-classical calculus

In this paper, we study the convergence of domain decomposition methods for the solving of advection-diffusion equations (0.1) (B del - h div C del)(u) = f, where B is a given vector field (B-x > 0), h is the viscosity and C is a positive definite symmetric matrix. Equation (0.1) models the transport of a quantity u (e.g, dyer, temperature, energy,...) by a vector field B and its diffusion scaled by a usually small viscosity coefficient h. It arises in many different areas like environmental flows, semiconductors, fluid dynamics,.... It is also involved in the numerical computation of Navier-Stokes solutions by successive linearizations techniques (see e.g. [10]). Domain decomposition methods are well-fitted to the solving of (0.1) for very large scale problems on parallel computers. Roughly speaking, the idea is to solve a boundary value problem by decomposing the domain into overlapping or nonoverlapping subdomains.