A simple, efficient, spectrally-accurate numerical method for solving variable-coefficient elliptic partial differential equations in periodic geometry is described. Numerical results show that the method is efficient and accurate even for difficult problems including convection-diffusion equations. Generalizations and applications to phase field models of crystal growth are discussed. Jb Received by the editors February 16, 1993. 1991 Mathematics Subject Classification. Primary 65N35, 65T20, 65N22.