stability of a FEM solution to NS equations

[popiol]

I'm looking for a numerical stability and error estimation of a finite element approximation of Navier-Stokes equations (with combustion). I define variables and operators on a domain that has both space and time axes (Ω = Ω_s x [0,t_max]), so the transport equation looks generally like this

 

div ( u [ v; 1 ] - D [ _su; 0 ] ) = Q,

 

where u is either density (of one of the species), velocity, temperature or pressure; v is velocity; D is diffusion coefficient; _s is gradient over the space domain Ω_s; and Q is either reaction rate, pressure gradient plus buoyancy force (-_sp + f_b), energy release or 0.

 

The approximation scheme is

 

u_i = Σ_j<i ( D ( d_ij^-1 - d_i.^-1) - [ v_j; 1 ] . ( x_j - x_i ) / d_ij ) w_ij u_j

 

where u_i is a value in the i-th mesh node; d_ij = ||x_j - x_i||; d_i.^-1= Σ_j<i w_ij / d_ij; Σ_j<i w_ij = 1; and x_i is a mesh node in Ω. Mesh nodes are sorted by time, so t(x_i) > t(x_j) => i > j.

 

It is a bit difficult to define stability in this case, but the following condition seems reasonable

 

lim _{i -> ∞} ( u_i - Σ_j<i u_jw_ij ) = 0,

 

which implies

 

lim _{i -> ∞}D ( d_ij^-1 - d_i.^-1) - [ v_j; 1 ] . ( x_j - x_i ) / d_ij = 1.

 

It should also be true that for each j > 0

 

Σ_i>j ( D ( d_ij^-1 - d_i.^-1) - [ v_j; 1 ] . ( x_j - x_i ) / d_ij ) w_ij = 1

 

Any idea how to verify those conditions?