An Efficient Spectral Method for Second-Order Elliptic Equations with Variable Coefficients on a Planar Circular Sector

This paper presents an efficient spectral method for solving second-order elliptic equations with variable coefficients defined on a planar circular sector. The main idea is to employ the domain mapping method to convert the original problem on a planar circular sector into an equivalent form on a standard rectangular domain. Appropriate weighted Sobolev spaces are then introduced to construct the weak formulation of the equivalent form and its discretization. The existence and uniqueness of the weak solution are rigorously established based on the Lax-Milgram lemma. Finally, some numerical experiments are presented, and the results demonstrate that the proposed algorithm is convergent and effective.