You should solve your problem on multiple, successively-refined meshes to show that the answer is not changing substantially. Many journals require this for any numerical results to demonstrate that the solution is converged.
Edited to add: See, for example, this PDF from the ASME Journal of Fluids Engineering which expresses their editorial policy on the control of numerical accuracy. Item 4 on page 1 speaks to mesh independence. This journal, and most ASME numerical journals, have a similar policy for submitted numerical results. Most of the points are excellent, though I don't necessarily agree with the requirement for second-order methods (#2).