An efficient Fourier-Laguerre spectral-Galerkin method for exterior problems of two-dimensional complex obstacles

Abstract

In this paper, we propose a Fourier-Laguerre spectral method for exterior problems of two-dimensional complex obstacles based on the mapping method. We first use a polar coordinate transformation to convert the exterior domains of complex obstacles into the exterior domain of the unit disk. Then we apply the polar coordinate transformation to the exterior problem of an elliptic equation, derive its weak formulation and prove the existence and uniqueness of the weak solution. On this basis, we construct the Fourier-Laguerre spectral Galerkin scheme, describe its numerical implementation and analyze the convergence of the numerical solution under the H1-norm. Numerical results indicate that our spectral Galerkin method is easy to implement and possesses high-order accuracy.