An Efficient Spectral Method for Second-Order Elliptic Equations with Variable Coefficients on a Planar Circular Sector
Qilong Zhu
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China.
Jihui Zheng *
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Spectral method, second-order elliptic equations, variable coefficients, planar circular sector, domain mapping, numerical experiments