An Efficient Spectral-Galerkin Method for Elliptic Equations in 2D Complex Geometries

A polar coordinate transformation is considered, which transforms the complex geometries into a unit disc. Some basic properties of the polar coordinate transformation are given. As applications, we consider the elliptic equation in two-dimensional complex geometries. The existence and uniqueness of the weak solution are proved, the Fourier–Legendre spectral-Galerkin scheme is constructed and the optimal convergence of numerical solutions under H -norm is analyzed. The proposed method is very effective and easy to implement for problems in 2D complex geometries. Numerical results are presented to demonstrate the high accuracy of our spectral-Galerkin method.