Abstract In this paper, we develop an efficient Fourier-Legendre spectral-Galerkin method for solving elliptic partial differential equations on general two-dimensional domains. A key core of our approach is employing a harmonic map to handle the general physical domains. This technique ensures broad geometric applicability, making the method highly effective for both complex star-shaped and nonstar-shaped domains. Moreover, this method is rigorously proved, with optimal convergence results established under H¹ H 1 -norm, which is independent of the domain boundary’s smoothness. The effectiveness and generality of the scheme are validated through some numerical examples on a wide variety of complex geometries.
Shi et al. (Tue,) studied this question.