提出了圆域上二阶变系数椭圆方程的一种有效的谱方法。首先，利用极坐标变换，将原问题转化为极坐标下的一种等价形式，根据极条件、边界条件以及θ方向的周期性，引入了适当的Sobolev空间，建立了极坐标系下二阶变系数椭圆方程的一种弱形式及其离散格式。然后，利用Lax-Milgram引理证明了弱解的存在唯一性。再由非一致带权Sobolev空间中投影算子的逼近性质和傅里叶基函数的逼近性质，证明了逼近解的误差估计。另外，将提出的算法延伸到奇异非线性二阶椭圆方程的计算，并给出了数值算例,数值结果表明该算法是收敛的和高精度的。

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.