Recently, in order to approximate the solution of a partial differential equation overa n irregular planar domains, several efficient and robust variational methods designed to generate smooth and convex grids on such regions have been proposed (3-9,11-14). For those grids, several schemes have also been designed, and for them it is quite clear how effortless the use of the grid logical rectangular data structure can be (1,2,15). This fact makes these schemes attractive competitors to the finite element methods, which use unstructured grids and, in consequence, non trivial data structures inorder to save the grid information. Nevertheless, one must acknowledge that, since triangulatiolns have been known for a while , finite element methods have been known for a while, finite element methods have been thoroughly studies, and there is a lot of research on how to assemble the systems required to solve a large class of equations. Thus, a question that arises in a natural way is how competitive are FE/FD methods, when applied to the structured convex grids generated for irregular regions-which often have elongated elements-, in order to produce the numerical solution in an easy computational way using structured grids and, at the same time, accurate enough by using finite elements. In this paper we show how to accomplish this goal, and a series of numerical examples at the end provided a good example of the validity of the approach.