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\(C^1\)-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

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Abstract

This paper is devoted to Professor Benyu Guo’s open question on the \(C^1\)-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the \(C^1\)-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories—interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global \(C^1\)-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the \(C^1\)-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov–Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a \(C^1\)-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

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Acknowledgements

Huiyuan Li: The research of this author is partially supported by the National Natural Science Foundation of China (NSFC 11871145). Weikun Shan: The research of this author is supported by the National Natural Science Foundation of China (NSFC 11801147) and the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2017QNJH20). Zhimin Zhang: The research of this author is supported in part by the National Natural Science Foundation of China (NSFC 11471031, NSFC 91430216, and NSAF U1530401) and the U.S. National Science Foundation (DMS-1419040).

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Authors and Affiliations

  1. State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Huiyuan Li

  2. School of Science, Henan University of Technology, Zhengzhou, 450001, Henan, China

    Weikun Shan

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

  4. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

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  1. Huiyuan Li
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  2. Weikun Shan
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Correspondence to Zhimin Zhang.

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Li, H., Shan, W. & Zhang, Z. \(C^1\)-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations. Commun. Appl. Math. Comput. 1, 403–434 (2019). https://doi.org/10.1007/s42967-019-00041-w

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  • DOI: https://doi.org/10.1007/s42967-019-00041-w

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