A new error analysis in finite element method, which is based on the element size in spatial directions respectively, was proposed. This analysis is different from the ordinary error analysis method, which is based on the overall size of elements. An error matching principle can be derived from the analysis presented here. The principle can be used in generating optimal grids and in analyzing the error of the calculation. A high precision finite element scheme was discussed upon the error matching principle, that uses the high order interpolation in the desired direction. The new error analysis method and the error matching principle were verified by the numerical tests. The results show that the proper grid sizes and right order of interpolation can minimize the error and improve the computational efficiency.