why are fine meshes unable to remove low-frequency error?
 

Why are fine meshes unable to remove low-frequency error? Can anyone explain the reason in a mathematical context?

The key is that any grid induces a projective cut-off at the wavenumber kc=pi/h (Nyquist). All wavenumbers greater than kc are eliminated by the discrete description. The components at wavenumbers k<kc must be filtered properly. But generally the filters have a transfer function that can smooth only the higher part of resolved wavenumbers (think about a function like sin(csi)/csi ). You see that low wavenumbers components are maintained almost unchanged.

Low-frequency means long-wavelength.

Once you discretize the equations you find that the solution at each local cell depends only on it and its nearest neighbors. The number of neighbors (i.e. nearest, nearest and next-to-nearest) depends on your choice of discretization. So the solution at each cell doesn't care what is the value/error of the cells far away from it. So there is a certain distance (i.e. spectral radius) up to which the value/errors in each cells influence one another.

As you make the mesh finer (i.e. smaller in mm), that physical distance/radius decreases.

The way to clean up these errors of course is to keep iterating and allow the errors to propagate throughout the domain and slowly get killed.