The problem of air circulation and heat transfer can be modeled through mass, momentum and energy conservation equations. Assuming incompressibility, the mathematical formulation for the general problem can be written as: Find u, p and в satisfying the following system, div (u) = 0, in Qx [0, T], (1)

Au

p — + p (Vu)u — 2p div s (u) + Vp + р^Р(в-в) = 0, in Q x [0,T], (2)

с)в

pcp — + pc^u. (V в) — к div Ve = 0, in Qx[0,T ] , (3)

Vu. n = 0 in Г„ x [0,T], u(x, t) = u(x, t) in Tux [0,T],u(x,0)=u0 in Q x [0,T], kVe. n = 0 in Гd x[0,T], в(х, t) = в in Г x[0,T] and в(х,0) = во(x), in Q

where: u = u(x, t) is the velocity vector, p=(x, t) is the pressure, в = в(х, t) is the temperature, p is the viscosity, p is the density, к is the thermal condutivity, в is the reference temperature, n is the normal vector, c is the

specific heat, p is the coefficient of thermal expansion, g is the gravity vector, e(u) = ^(Vu + Vur), Q is the bounded domain with boundary Г = Гц иГ, = Гс иrd with Гц пГ, = Гс пГd =0 and the time t є [0,T].

The term pgp(e — воо ) allows the coupling of the air circulation and the heat transfer problems.

2. Methods

For the air circulation problem the numerical solutions are here obtained by a stabilized mixed finite element method that allows us to deal with the difficulties that come from the first equation system, Equations (1) and (2): the difficulty in constructing approximation spaces for problems with internal constraint; non-linearities of the convective type and numerical instabilities when advection effects are dominant. Here, a Petrov-Galerkin type method [6] was implemented and applied to analyze indoor air circulation cases, ensuring stability for dominant advection and for the internal constraint. In the case

of a heat transfer problem a stabilized finite element method was implemented — Streamline Up-wind Petrov-Galerkin (SUPG) [7].

Being L and H1 the usual Hilbert spaces and Rf the Lagrange polynomial space of the degree l and class C0. Then, defining the following approximation spaces

V={u„ є (H0Q)nRf))2,u„(x, t) = Uh(x, t) in Г, }c (HW, V0={v„ є (H0Q)nR(Q))2,v„(x, t) = 0 in f }c (Hf))2, Ph=[pf є (L2(Q) n Rf (Q)); I PfdQ = 0jc (L2(Q)), Sh = {% % (x, t) є(н1 (Q)n Rf (Q)) ,% (x, t) = % (x, t) in Гс }c (H*(Q)), Sf = {s„ sh (x, t) є (H1 (Q)n Rf (Q)), Sf (x, t) = 0 in Гс j c (H *(Q)) with the usual norm ||u||2 = | |u 110+lVull0 of Hl and N HIp0of L2.

The wind field can be determined by solving the following formulation:

Find {uh, ph } є Vh x Ph satisfying the following system B (uf, Ph; vh, q) = ° v (vh, q) єКx Ph, where:

((Vvh ) ah — 2^ div Фи ) + V4h ))h+H ph, 4h ), v vh є Vh e Чи є Ph.

with у << 1 and and Д stabilized parameters suggested by Franca and Frey [6].

And find %(x, t) satisfying the following system:

With r= kuVs^,the SUPG stabilized parameters suggested by Brooks and Hughes [7]. The time discretization has been done by backward Euler finite differences.