From the theoretical approach to the fill-in minimization problem we present one of the optimal vertex elimination process for a regular finite element mesh M (nxn), and through a number of numerical experiments it is verified that the new process model can always lead to better numerical results comparing to other methods presently in use. Since the process here presented cann't give the actual dissections of M but can clarify how the optimal elimination is, the process includes George's Nested Dissection Method and the method by Duff, Erisman and Reid. By this investigation we can conclude that l) the concept of "Dissection" is neccessary for minimizing the number of fill-ins, 2) the location of the dissection lines can be systematically decided even if n of M is odd or even number, and though the interior area of M is dissected as George's Method, the surrounding area of M is rather irregularily dissected, and 3) the model of the vertex elimination process given in this paper is applied to other kind of regular finite element mesh or finite difference mesh, too.