A lattice Boltzmann method is developed for modeling viscous elementary flows. An adjustable source term is added to the lattice Boltzmann equation, which can be tuned to model different elementary flow features like a doublet or a point source of any strength, including a negative source (sink). The added source term is dimensionally consistent with the lattice Boltzmann equation. The proposed model has many practical applications, as it can be used in the framework of the potential flow theory of viscous and viscoelastic fluids. The model can be easily extended to the three dimensional case. The model is verified by comparing its results with the analytical solution for some benchmark problems. The results are in good agreement with the analytical solution of the potential flow theory.