A Multidomain Decomposition for the Transport Equation

Abstract

In this presentation we will discuss a domain decomposition technique for advection equations of the following type:

$$ \frac{{\partial u}}{{\partial t}} + divb\left( u \right) = f, $$

(1.1)

where f is given and b(u) is a transport field possibly depending upon the unknown u. This equation has to be fulfilled in a spatial region and suitable conditions at the boundary of this region and at the initial time must be prescribed. As a matter of fact, our analysis will be carried out on a linearized, time independent version of equation (1.1) (to this case one can always reduce after a time discretization and a suitable linearization of the nonlinear term b(u)). More precisely, we shall consider the following boundary value problem

$$ \left\{ {\begin{array}{*{20}{c}} {div\left( {bu} \right) + {b_0}u = f in \Omega ,}\\ {u = g on \partial {\Omega ^{in}},} \end{array}} \right. $$

(1.2)

where Ω is a two dimensional domain, b, f and b ₀ are assigned functions in Ω and g is a given function defined in the portion ∂Ω ⁱⁿ of the boundary of Ω along which the transport enters Ω (∂Ωn ⁱⁿ is said to be the “inflow boundary”).

The following institutions have provided a partial support for this work: Istituto di Analisi Numerica del Consiglio Nazionale delle Ricerche (Pavia, Italy), Ministero dell’Università e della Ricerca Scientifica e Tecnologica (Italy), Institute for Mathematics and its Applications (funds provided by the National Science Foundation).