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 (\(\Omega = \Omega_s \times [0,t_{\max}]\)), so the transport equation looks generally like this div ( \(u [ v; 1 ] - D [ \nabla_s u; 0 ]\) ) = Q, where \(u\) is either density (of one of the species), velocity, temperature or pressure; \(v\) is velocity; \(D\) is diffusion coefficient; \(\nabla_s\) is gradient over the space domain \(\Omega_s\); and Q is either reaction rate, pressure gradient plus buoyancy force (-\(\nabla\)p + \(f_b\)), energy release or 0. The approximation scheme is \(u_i = \sum_{j<i} ( D ( d_{ij}^{-1} - d_{i.}^{-1}) - [ v_j; 1 ] \cdot ( x_j - x_i ) / d_{ij} ) w_{ij} u_j\) where \(u_i\) is a value in the \(i\)-th mesh node; \(d_{ij} = \|x_j - x_i\|\); \(d_{i.}^{-1} = \sum_{j<i} w_{ij} / d_{ij}\); \(\sum_{j<i} w_{ij} = 1\); and \(x_i\) is a mesh node in Ω. Mesh nodes are sorted by time, so \(t(x_i) > t(x_j) \Rightarrow i > j\). It is a bit difficult to define stability in this case, but the following condition seems reasonable \(\lim_{i \rightarrow \infty} ( u_i - \sum_{j<i} u_j w_{ij} ) = 0\), which implies \(\lim_{i \rightarrow \infty} D ( d_{ij}^{-1} - d_{i.}^{-1}) - [ v_j; 1 ] \cdot ( x_j - x_i ) / d_{ij} = 1\). It should also be true that for each \(j > 0\) \(\sum_{i>j} ( D ( d_{ij}^{-1} - d_{i.}^{-1}) - [ v_j; 1 ] \cdot ( x_j - x_i ) / d_{ij} ) w_{ij} = 1\) Any idea how to verify those conditions?

Why do you use that approximation scheme? You said you are using finite elements, but what you wrote would be a finite difference scheme. What is the motivation behind defining stability in this way? Why do you say it is reasonable? Why it is that "it should also be true" on the last condition?

Finite element stuff isn't my thing but surely the numerical stability of the NS equations in fluid mechanics must literally fill libraries? There's tons of grid generation and solving textbooks out there. Are you trying to come up with your own or is this a well known scheme but you just want to see how to check it's reasonable?