Convergence In the Poiseuille Equation

Consider a mesh as follows (Which is a unit square):

I solved the Poiseuille equation by using the exact same code as mentioned in the tutorial

Then, I chose 100 points spread uniformly across this unit square (Need not to be the grid points). Let’s call them as X_{grid}.

After that I evaluated the velocity values in this X_{grid}. Let’s denote these set evaluated velocity values as V^1_{grid}.

Next, I reduced the mesh size, Solved the Poiseuille equation and again evaluated the velocity values in the X_{grid} points. Call it V^2_{grid}

I continued to do reduce the mesh sizes and evaluated the velocity values in the same X_{grid} points. That is: V^3_{grid}, V^4_{grid}, V^5_{grid}

So, technically as I’m refining the mesh, the velocity values should be closer to the exact answer in the latter solutions.

However, when I evaluate the exact solution in the X_{grid}, call it as V_E and the difference was taken, That is:
||V^1_{grid}-V_E||, ||V^2_{grid}-V_E||, ||V^3_{grid}-V_E||, ||V^4_{grid}-V_E||, ||V^5_{grid}-V_E||, ||V^6_{grid}-V_E||, ||V^7_{grid}-V_E||, ||V^8_{grid}-V_E||

The answer looks like this:

Which shows that the finest mesh has a larger error.

May I know whether something like that is at all possible? (May be the tiny meshes has caused lot of numerical calculations?)

The reason for not posting my code is because, it is kind of long and I cannot think of a way to simplify it. But if it is really necessary, I can show that as well.