Volume 12, Issue 3
A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain

Adv. Appl. Math. Mech., 12 (2020), pp. 815-834.

Published online: 2020-04

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• Abstract

In this paper, an efficient spectral method is applied to solve fourth order elliptic eigenvalue problems in circular domain. Firstly, we derive the essential pole conditions and the equivalent dimension reduction schemes of the original problem. Then according to the pole conditions, we define the corresponding weighted Sobolev spaces. Together with the minimax principle and approximation properties of orthogonal polynomials, the error estimates of approximate eigenvalues are proved. Thirdly, we construct an appropriate set of base functions contained in approximation spaces and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate the theoretical results and algorithms.

• Keywords

Fourth order elliptic eigenvalue problems, dimension reduction scheme, error analysis, numerical algorithms, circular domain.

65N30, 65N35

g15725795176@163.com (Yixiao Ge)

yezitanting@163.com (Ting Tan)

aj154@163.com (Jing An)

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@Article{AAMM-12-815, author = {Ge , Yixiao and Tan , Ting and An , Jing}, title = {A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {3}, pages = {815--834}, abstract = {

In this paper, an efficient spectral method is applied to solve fourth order elliptic eigenvalue problems in circular domain. Firstly, we derive the essential pole conditions and the equivalent dimension reduction schemes of the original problem. Then according to the pole conditions, we define the corresponding weighted Sobolev spaces. Together with the minimax principle and approximation properties of orthogonal polynomials, the error estimates of approximate eigenvalues are proved. Thirdly, we construct an appropriate set of base functions contained in approximation spaces and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate the theoretical results and algorithms.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0068}, url = {http://global-sci.org/intro/article_detail/aamm/16425.html} }
TY - JOUR T1 - A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain AU - Ge , Yixiao AU - Tan , Ting AU - An , Jing JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 815 EP - 834 PY - 2020 DA - 2020/04 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0068 UR - https://global-sci.org/intro/article_detail/aamm/16425.html KW - Fourth order elliptic eigenvalue problems, dimension reduction scheme, error analysis, numerical algorithms, circular domain. AB -

In this paper, an efficient spectral method is applied to solve fourth order elliptic eigenvalue problems in circular domain. Firstly, we derive the essential pole conditions and the equivalent dimension reduction schemes of the original problem. Then according to the pole conditions, we define the corresponding weighted Sobolev spaces. Together with the minimax principle and approximation properties of orthogonal polynomials, the error estimates of approximate eigenvalues are proved. Thirdly, we construct an appropriate set of base functions contained in approximation spaces and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate the theoretical results and algorithms.

Yixiao Ge, Ting Tan & Jing An. (2020). A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain. Advances in Applied Mathematics and Mechanics. 12 (3). 815-834. doi:10.4208/aamm.OA-2019-0068
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