An efficient spectral method for second order elliptic equation with variable coefficient on a circular domain is proposed. At first, the original problem is transformed into an equivalent form in polar coordinate by using the polar coordinate transformation. Then according to the polar condition, boundary condition and the periodicity in θ direction, some appropriate Sobolev spaces are introduced, and a weak form and its discrete scheme are derived.Based on Lax-Milgram lemma, the existence and uniqueness of the weak solution are proved. In addition, from the approximation property of Fourier basis function and projection operator, the error estimation of the approximation solution is proved.Moreover, the algorithm is extended to the singular nonlinear second order elliptic equation. Some numerical examples are presented, and numerical results show that the algorithm is convergent and high-accuracy.