In the end, the error depends on *all* parameters including the inlet concentration profile.

It even depends on the norm in which it is measured.

From *a priori* convergence analysis, one often obtains estimates like

\lVert c - c_{h}\rVert \leq C h^k, \label{eq:error-apriori} \tag{E}

where c is the true solution of the model, c_h is the numerical approximation, h is a discretization parameter (i.e., the finite volume cell size, finite element size, finite difference step size), k is the order of convergence, and C > 0 is some constant.

As mentioned before, the constant C and the order k depend on the norm \lVert \cdot \rVert (e.g., L_2, L_\infty) in general. It is also important *where* the error is measured (e.g., only at the outlet, over the full column length, including particles, only at the end time etc.).

It’s even more complicated: The constant C and the order k depend on the solution c itself. The maximal order is typically only reached for smooth solutions c.

Long story short: We don’t know C and can only compute it for a specific problem (i.e., specific set of parameters with that specific inlet profile). Without knowing C, it is impossible to determine h such that the error is lower than a given tolerance (using a priori estimates of this type).

How to compute k and C for some specific problem: We require an analytical solution (or a very precise reference solution). We then compute an error plot by evaluating \eqref{eq:error-apriori} for many values of h that get smaller and smaller (e.g., h = 10^{-i}, i = 1,2, \dots). This way, we make sure to reach the asymptotic regime, where the error is proportional to h^k. The constants k and C are then estimated from the plot (using only datapoints from the asymptotic regime) by regression.

Error estimates that are more precise than \eqref{eq:error-apriori} are often given by *a posteriori* error estimators. However, these may also contain constants, which are difficult to compute and you have to solve the problem at least once.