Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional （2-D） and three dimensional （3-D） space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.
