The electronic structures and dynamics in atoms and molecules are governed by the Coulomb potential. For numerical simulations, it requires to treat the short- and long-range behaviors of the Coulomb potential with dense local refinement and large spatial dimension. To attack the Coulomb potential in all-electron calculations of polyatomic molecules, I have developed new numerical grid-based methods: Voronoi-cell finite difference (VFD) method and time-dependent VFD method. In contrast to the ordinary finite difference method with regular uniform grids, the VFD method can accommodate any types of grid distributions, so called unstructured grids, with the help of geometrical adaptivity of the Voronoi diagram, while it retains simplicity of the finite difference method. Non-uniform and highly adaptive multicenter molecular grids are constructed by a combination of spherical atomic grids centered at nuclear positions. The new methods have been applied to solve density-functional theory (DFT) [1] and time-dependent DFT (TDDFT) [2] equations for accurate electronic structure and dynamics calculations of polyatomic molecules on multicenter molecular grids.