An efficient and accurate mapping method for elliptic equations in irregular annular domains

In this paper, we introduce a coordinate transformation, which transforms the irregular annular domain to a unit disk. We present its basic properties. As examples, we consider Poisson type equation and Cauchy–Navier elastic equations with variable coefficients in two-dimensional irregular annular domains, and prove the existence and uniqueness of weak solutions. We also construct the mixed Fourier–Legendre spectral schemes, and derive the optimal convergence of numerical solutions under the $H^1$-norm. The numerical results indicate that the suggested method achieves high-order accuracy.