What is the linear preserving criterion? I have a question regarding the linearity preserving criterion:

From an article I read, is said that the linearity preserving criterion required that the discretization scheme is exact whenever the solution is piecewise linear and the diffusion tensor is piecewise constant. 

When writing "exact", does this mean that the scheme is an exact representation of the real world, or is this referring to something else?

When writing that it is exact whenever the solution is piecewise linear, does this mean that the solution gives is piecewise linear, it follows that this solution is exact?

I understant the linearity preserving criterion is very important when we have distorted mesh, but as you can see I have a problem understanding what the linearity preserving criterion actually is.
It means that you are assuming that for a given degree of freedon on your mesh, the diffusion phenomenon behaves linearly. In other words, for a 1D problem, Integral of Grad(F) = 0 will give you a straight line. This will be valid for each degree of freedom, since you are assuming that your physics behaves linearly at that point. Wich is fairly reasonable. If you zoom in the continuum, the solution, if your derivatives are continuous, then you´d see a straight line for a delta x as small as possible.

I hope it will help.
It means that you are assuming that for a given degree of freedon on your mesh, the diffusion phenomenon behaves linearly. In other words, for a 1D problem, Integral of Grad(F) = 0 will give you a straight line. This will be valid for each degree of freedom, since you are assuming that your physics behaves linearly at that point. Wich is fairly reasonable. If you zoom in the continuum, the solution, if your derivatives are continuous, then you´d see a straight line for a delta x as small as possible.

I hope it will help.