A new Galerkin spectral element method for fourth-order boundary value problems

Abstract

In this paper, we propose a new Galerkin spectral element method for one-dimensional fourth-order boundary value problems. We first introduce some quasi-orthogonal approximations in one dimension, and establish a series of results on these approximations, which serve as powerful tools in the spectral element method. By applying these results to the fourth-order boundary value problems, we establish sharp $H^2$ and $L^2$ error bounds of the Galerkin spectral element method. The efficient algorithm is implemented in detail. Numerical results demonstrate its high accuracy, and confirm the theoretical analysis well.

Keywords:

• fourth-order boundary value problems
• Galerkin spectral element method
• quasi-orthogonal approximations

2000 Mathematics Subject Classifications:

• 65N35
• 35J40
• 41A10

Disclosure statement

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

Additional information

Funding

The research of the first author is supported in part by the NSF of China Nos. 11326244, 11401380, 11301338 and the Fund for Young Teachers of Shanghai Universities [No. ZZshjr12009]. The research of the second author is supported in part by the NSF of China [No. 11301343], the NSF of Shanghai [No. 15ZR1430900] and the Fund for E-institute of Shanghai Universities [No. E03004].