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 H1-norm. The numerical results indicate that the suggested method achieves high-order accuracy.