Several examples showing the reliability and accuracy of finite element/difference methods using standard structured convex grids over very irregular regions, generated by the direct variational method, have been recently reported. In this paper, we present a survey of the application of both methods to some test problems, for which we have selected quality structured grids generated by minimizing suitable smoothness and area functionals, and show the accuracy improvement of the results on such grids. As a conclusion, we propose a geometric adaptive strategy for generating useful grids on very irregular regions.