# Stability of a FEM solution to NS equations

1. Mar 9, 2012

### 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 = Ωs x [0,tmax]), so the transport equation looks generally like this

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

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

The approximation scheme is

ui = Σj<i ( D ( dij-1 - di.-1) - [ vj; 1 ] . ( xj - xi ) / dij ) wij uj

where ui is a value in the i-th mesh node; dij = ||xj - xi||; di.-1= Σj<i wij / dij; Σj<i wij = 1; and xi is a mesh node in Ω. Mesh nodes are sorted by time, so t(xi) > t(xj) => i > j.

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

lim i -> ∞ ( ui - Σj<i uj wij ) = 0,

which implies

lim i -> ∞ D ( dij-1 - di.-1) - [ vj; 1 ] . ( xj - xi ) / dij = 1.

It should also be true that for each j > 0

Σi>j ( D ( dij-1 - di.-1) - [ vj; 1 ] . ( xj - xi ) / dij ) wij = 1

Any idea how to verify those conditions?