Sobolev orthogonal Legendre rational spectral methods for exterior problems

Abstract

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.

Keywords:

• Modified Legendre rational functions
• spectral methods
• Sobolev orthogonal functions
• exterior problems
• numerical results

2000 Mathematics Subject Classifications:

• 65N35
• 41A20
• 33C45
• 35J25
• 35K10

Acknowledgements

The work is supported by the National Natural Science Foundation of China (No. 12071294).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the National Natural Science Foundation of China (No. 12071294).