Laterally periodic resonators, which can be constructed by use of transversely periodic phase- or amplitude-modulating elements in a cavity, are proposed for stabilization and generation of transversely coherent output from large-area gain. Lasers with periodic resonators have the combined features of conventional cavities and laser arrays. Significant low-order transverse modes and mode discrimination of a sample resonator with intracavity periodic phase elements are investigated numerically by the iteration method. Wave-propagation calculations are carried out by use of a fast Fourier transform, and a modified Prony method is used to evaluate wave functions and losses of transverse modes. Results of numerical calculations are consistent with expectations.