The electronic structure calculation of atoms is one of most fundamental bases in not only understanding electronic structures of molecules and solids, but also developing efficient and accurate electronic structure methods. For the latter case, a highly accurate method should be used to distinguish the intrinsic error produced by the theoretical framework itself from that caused by the other numerical problems such as incompleteness of basis set and inaccurate numerical integration. As a such method, we developed an accurate finite element method for atomic calculations based on density functional theory (DFT) within local density approximation (LDA) and Hartree-Fock (HF) method. The radial wave functions are expanded by cubic Hermite spline functions on a uniform mesh for x=√ r , and all the associated integrals are analytically evaluated in conjunction with fitting procedures of the hartree and the exchange-correlation potentials to the same cubic Hermite spline functions using a set of recurrence formulas. The total energy of atoms systematically converges from above, and the error algebraically decays as the mesh spacing decreases. When the mesh spacing is taken to be 0.025/√ Z bohr, the total energy for an atom of atomic number Z can be calculated within error of 10-7 hartree for both the LDA and HF methods. The equal applicability of the method to DFT and the HF method with a similar degree of high accuracy enables the method to be a reliable platform for development of new functionals in DFT such as hybrid functionals.