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(adsbygoogle = window.adsbygoogle || []).push({}); _{s}x [0,t_{max}]), so the transport equation looks generally like this

div (u[v; 1 ] -D[ grad_{s}u; 0 ] ) =Q,

whereuis either density (of one of the species), velocity, temperature or pressure;vis velocity;Dis diffusion coefficient; grad_{s}is gradient over the space domain Ω_{s}; andQis either reaction rate, pressure gradient plus buoyancy force (-grad_{s}p + 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}

whereu_{i}is a value in thei-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; andx_{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_{j}w_{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 eachj> 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Stability of a FEM solution to NS equations

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**