The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy.

本文的目的是发展外部问题的对角化勒让德有理谱方法。我们首先考虑极坐标下二维椭圆和抛物方程的外部问题，构造Sobolev正交Legendre有理基函数，然后提出对角化Legendre有理谱方法。然后考虑球坐标下三维椭圆和抛物方程的外部问题，构造Sobolev正交Legendre有理基函数，进而提出对角化Legendre有理谱方法。建议方法的主要优点是离散系统是对角的，并且数值解可以表示为截断的傅里叶级数。数值结果表明了它们的有效性和准确性。