Optimal control of nonlinear partial differential equations (PDEs) is an open problem with applications that include fluid, thermal, biological, and chemically-reacting systems. Receding horizon control with the continuation/ generalized minimum residual (C/GMRES) method is a fast algorithm to solve the optimal control problem of nonlinear systems described by ordinary differential equations. In this paper, we develop a design method of the receding horizon control for nonlinear systems described by partial differential equations. Our approach is a direct infinite dimensional extension of the receding horizon control method for finite-dimensional systems. In this paper, we moreover propose an efficient algorithm rather than the C/GMRES algorithm for numerically solving the nonlinear receding horizon control problem. The effectiveness of the proposed method is verified by numerical simulations.