The discrete singular convolution (DSC) algorithm, a new real-space mesh technique proposed in recent years, has been mostly applied in fields of mathematics, physics and engineering, and rather positive responses have been received compared with other related methods. It can be much more efficient and accurate in many cases. Such positive responses encourage us to explore its applications in physics and materials science, with a long-range goal in ab initio calculations. The application of the DSC algorithm in solving the Schrödinger equation of one-electron systems was attempted with a hydrogen atom as an example. Using the uniform discretization and Shannon kernel, the Schrödinger equations were solved in spherical coordinates. Compared with other methods such as discrete variable representation, Hartree-Fock, and density functional theory, the DSC algorithm is robust and efficient in numerical solutions for achieving accurate eigenvalues of excited states, and thus, especially suitable for problems in which lots of accurate eigenvalues of excited states are requested. The restrictions of boundary conditions on the discretization and the impact of singularities were studied. Both the boundary conditions and singularities were found to be critical in the numerical applications. Then we generalized the DSC algorithm from uniform discretization to non-uniform discretization by introducing a mapping method to regularize the singularities involved. The approach was demonstrated using a radial Schrödinger equation of a hydrogen atom with a Coulomb-like potential that involves a singularity. The Schrödinger equation of a one-dimensional (1D) hydrogen atom was also solved by such an approach. The new approach provides accurate eigenenergies at much reduced computational cost. The mapping method has advantages over the conventional regularization method of the discrete variable representation (DVR) and its descendant, the Lagrange-mesh method in that it can be easily realized in computations and is convenient for selecting suitable grid points. We also developed a second mapping method to make DSC adaptive to Coulomb singularity. In the first mapping method, the basis functions in the uniform grid space are mapped to a non-uniform grid space, and we use the new basis functions in the non-uniform grid space to construct the delta-type DSC kernel. However, in the second mapping method, we construct the delta-type DSC kernel in the uniform grid space first, and then map such original kernel to the non-uniform grid space to obtain its non-uniform version directly. We applied the second mapping method to the radial hydrogen atom and the 1D hydrogen atom too and obtained similar results. We also demonstrated the connection between the DSC method and the DVR method.