An Effective Spectral Method For Second Order Elliptic Equations With Variable Coefficients In Polar Geometry And Its Application

In this paper,we propose an efficient spectral method for second order elliptic equation with variable coefficient in the circular domain and circular ring.At first,the original problem is transformed into an equivalent form in polar coordinate by using the polar coordinate transformation for the circular domain.Then according to the polar condition,boundary condition and the periodicity in direction,we introduce some appropriate Sobolev spaces,and derive a weak form and its discrete scheme.Based on Lax-Milgram lemma,we prove the existence and uniqueness of the weak solution.In addition,from the approximation property of Fourier basis function and projection operator,we prove the error estimation of the approximation solution.Moreover,we extend our algorithm to the singular nonlinear second order elliptic equation.We present some numerical examples,and the numerical results show that our algorithm is convergent and high-accuracy.For elliptic equation with variable coefficient on a circular ring,we propose an efficient spectral method.At first,the original problem is transformed into an equivalent form in polar coordinate by using the polar coordinate transformation.Then according to the boundary condition and the periodicity in direction,we introduce some appropriate Sobolev spaces,and derive a weak form and the corresponding discrete scheme.Finally,we provide some numerical experiments,and numerical results show the efficiency of our algorithm.