Galerkin–Legendre spectral method for Neumann boundary value problems in three dimensions

ABSTRACT

In this paper, we investigate the Legendre spectral methods for problems with the essential imposition of Neumann boundary condition in three dimensions. A double diagonalization process has been employed, instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of Neumann boundary condition. For analysing numerical errors, some results on Legendre orthogonal approximation in Jacobi weighted Sobolev space are established. As examples of applications, the spectral schemes are provided for two model problems. The convergences of the proposed schemes are proved, too. Numerical results demonstrate the spectral accuracy in space, and which confirm theoretical analysis well.

KEYWORDS:

• Galerkin–Legendre spectral method
• Neumann boundary value problems
• Legendre approximation in Jacobi weighted Sobolev space
• elliptic equations in three dimensions
• explicit lifting technique

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N35
• 65N12
• 42C10
• 35J25

Acknowledgments

The authors would like to thank Associate Professor Li Wei of Henan University of Science and Technology and Doctor Wang Lu-yu of Zhejiang University for their suggestions to improve the quality of the paper. We also would like to thank the anonymous referees for their valuable suggestions and comments, which greatly helped improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Tian-jun Wang is supported in part by NSF of China [grant numbers 11371123, 11571151 and 11771299]. The work of Tao Sun is supported in part by NSF of China [grant numbers 11401380, 11671166 and 11701371] and first level discipline project funding in Statistics subject of Shanghai Lixin University of Accounting and Finance [grant number 2017-10].