The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem

Jinhua Feng; Shixi Wang; Hai Bi; Yidu Yang; Jinhua Feng; Shixi Wang; Hai Bi; Yidu Yang

doi:10.3934/math.2024163

AIMS Mathematics

2024, Volume 9, Issue 2: 3332-3348. doi: 10.3934/math.2024163

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The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem

1.
Qiushi College, Guizhou Normal University, Guiyang, Guizhou 550025, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China

Received: 27 September 2023 Revised: 03 December 2023 Accepted: 20 December 2023 Published: 04 January 2024
MSC : 65N25, 65N30

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [Numer. Methods Partial Differ. Equ., 27 (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy.
- the biharmonic eigenvalue,
- Ciarlet-Raviart mixed method,
- conforming finite element,
- a posteriori error estimator,
- adaptive algorithm
Citation: Jinhua Feng, Shixi Wang, Hai Bi, Yidu Yang. The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem[J]. AIMS Mathematics, 2024, 9(2): 3332-3348. doi: 10.3934/math.2024163

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Abstract

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [Numer. Methods Partial Differ. Equ., 27 (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy.

References

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