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 [ grad_su; 0 ] ) = Q,

where u is either density (of one of the species), velocity, temperature or pressure; v is velocity; D is diffusion coefficient; grad_s is gradient over the space domain Ω_s; and Q is either reaction rate, pressure gradient plus buoyancy force (-grad_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_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 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?

 

[jvicens]

I would first commend you on your thorough and well-defined approximation scheme. It appears that you have considered all the necessary variables and operators in your transport equation and have taken into account the space and time axes of your domain.

To address your question about numerical stability and error estimation, it is important to first define what we mean by "stability" in this context. In general, stability refers to the ability of a numerical method to produce accurate results in the presence of small perturbations or errors. In the case of finite element approximation of Navier-Stokes equations, stability can refer to both the stability of the scheme itself (i.e. the accuracy of the computed solution) and the stability of the numerical implementation (i.e. the ability of the scheme to handle errors and perturbations).

In order to verify the conditions you have outlined, it would be helpful to perform a stability analysis of your scheme. This involves analyzing the behavior of your scheme as the mesh size and time step are varied. In particular, you will want to investigate how the error in your solution (i.e. the difference between the exact solution and the numerical solution) changes as the mesh size and time step are refined. This will allow you to determine the convergence rate of your scheme and assess its stability.

Another important aspect to consider is the error estimation in your scheme. This involves quantifying the error in your numerical solution and determining its sources. In the case of finite element approximation, the error can arise from various sources such as numerical discretization, truncation error, and round-off error. By understanding the sources of error, you can then take steps to minimize them and improve the accuracy of your solution.

In summary, to verify the conditions you have outlined, I would recommend performing a stability analysis and error estimation of your finite element approximation scheme. This will allow you to determine the convergence rate and stability of your scheme, as well as identify any sources of error that may be affecting the accuracy of your solution.