Reduction in Residual Norm vs. Mesh Refinement

CFD Rookie

Hello everyone, I kinda in trouble again. Let's say that I have an analysis (BC is fixed). I first start with a coarse mesh and reach a converged solution. By looking at the residual norm, I am not satisfied, the residuals still look too big and it is not reducing any further. So the second step is I refine my mesh, especially at those locations where there are hugh pressure and velocity changes (gradients). Then based on this finer mesh, I re-run the analysis (to save time, I use the final result of the 1st run as my re-starting point). So at the end of the second run, should I DEFINITELY expect the norm residuals to be samller than those in the run #1?

Thanks in advance.