Adaptive mesh generation using a posteriori error estimation has recently attracted great interest. This paper proposes a new r-method of adaptive mesh generation for the finite element method. The objective function of this method is to minimize the error norm of the whole computational domain. It is very impotant for the adaptive mesh generation to estimate the numerical error accurately. The discretization error in the finite element method is estimated using the computer code ERRAND-FEW developed by the authors. In order to verify the newly proposed adaptive method, we have chosen two 2-dimensional stress concentration and singularity problems. The results show the validity and the effectiveness of this method. Since this method is very simple and flexible, it is quite possible to extend it to any other problems, such as shell problems or solid problems.