C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

doi:10.1007/s42967-019-00041-w

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (3): 403-434.doi: 10.1007/s42967-019-00041-w

• ORIGINAL PAPER • 上一篇下一篇

C¹-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

Huiyuan Li¹, Weikun Shan², Zhimin Zhang^3,4

1 State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China;
2 School of Science, Henan University of Technology, Zhengzhou 450001, Henan, China;
3 Beijing Computational Science Research Center, Beijing 100193, China;
4 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

收稿日期:2018-08-16 修回日期:2019-04-19 出版日期:2019-09-20 发布日期:2019-09-09
通讯作者: Zhimin Zhang E-mail:zmzhang@csrc.ac.cn
基金资助:
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).

C¹-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

Huiyuan Li¹, Weikun Shan², Zhimin Zhang^3,4

1 State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China;
2 School of Science, Henan University of Technology, Zhengzhou 450001, Henan, China;
3 Beijing Computational Science Research Center, Beijing 100193, China;
4 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Received:2018-08-16 Revised:2019-04-19 Online:2019-09-20 Published:2019-09-09
Supported by:
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).

摘要/Abstract

摘要： This paper is devoted to Professor Benyu Guo's open question on the C¹-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 C¹-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 C¹-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 C¹-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 C¹-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.

关键词: Quadrilateral spectral element method, Fourth-order equations, Mapped polynomials, C¹-conforming basis, Polynomial inclusion, Completeness

Abstract: This paper is devoted to Professor Benyu Guo's open question on the C¹-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 C¹-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 C¹-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 C¹-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 C¹-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.

Key words: Quadrilateral spectral element method, Fourth-order equations, Mapped polynomials, C¹-conforming basis, Polynomial inclusion, Completeness

中图分类号:

65N35
41A10

Huiyuan Li, Weikun Shan, Zhimin Zhang. C¹-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 403-434.

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C¹-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

C¹-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

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