A new finite element grid generation technique using Delaunay triangulation is presented. Delaunay triangulation is known best triangulation of arbitrarily given points. A set of points are given on the boundary of the physical domain under consideration before initial triangular elements are first generated. The 'quality' of every element is then calculated to create new interior points near the elements of poor 'quality'. The density of elements can be controlled dually by initial boundary points and background grid information. To validate the performance of the present grid system, Euler equations have been numerically solved for two test problems by Taylor-Galerkin Scheme with flow-adaptive grid capability. Transonic external flow over NACA0012 airfoil with one degree of angle of attack was the first problem and supersonic internal flow over 4\% arc bump in a channel was the second.
