Volume 12, Issue 3
A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem

Adv. Appl. Math. Mech., 12 (2020), pp. 774-796.

Published online: 2020-04

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• Abstract

In this paper, a type of cascadic adaptive finite element method is proposed for eigenvalue problem based on the complementary approach. In this new scheme, instead of solving the eigenvalue problem in each adaptive finite element space directly, we only need to do some smoothing steps for a boundary value problems on each adaptive space and solve some eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the eigenvalue problems on each adaptive space which is time-consuming. Further, the complementary error estimate for eigenvalue problem will be introduced. This estimate can not only provide an accurate error estimate for eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.

• Keywords

Adaptive finite element method, cascadic multigrid method, eigenvalue problem, complementary method.

• AMS Subject Headings

65N30, 35J61, 65M55, 65B99

xufei@lsec.cc.ac.cn (Fei Xu)

qmhuang@bjut.edu.cn (Qiumei Huang)

chenshuangshuang@bjut.edu.cn (Shuangshuang Chen)

mahongk@mail.sysu.edu.cn (Hongkun Ma)

• BibTex
• RIS
• TXT
@Article{AAMM-12-774, author = {Xu , Fei and Huang , Qiumei and Chen , Shuangshuang and Ma , Hongkun}, title = {A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {3}, pages = {774--796}, abstract = {

In this paper, a type of cascadic adaptive finite element method is proposed for eigenvalue problem based on the complementary approach. In this new scheme, instead of solving the eigenvalue problem in each adaptive finite element space directly, we only need to do some smoothing steps for a boundary value problems on each adaptive space and solve some eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the eigenvalue problems on each adaptive space which is time-consuming. Further, the complementary error estimate for eigenvalue problem will be introduced. This estimate can not only provide an accurate error estimate for eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0054}, url = {http://global-sci.org/intro/article_detail/aamm/16423.html} }
TY - JOUR T1 - A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem AU - Xu , Fei AU - Huang , Qiumei AU - Chen , Shuangshuang AU - Ma , Hongkun JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 774 EP - 796 PY - 2020 DA - 2020/04 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0054 UR - https://global-sci.org/intro/article_detail/aamm/16423.html KW - Adaptive finite element method, cascadic multigrid method, eigenvalue problem, complementary method. AB -

In this paper, a type of cascadic adaptive finite element method is proposed for eigenvalue problem based on the complementary approach. In this new scheme, instead of solving the eigenvalue problem in each adaptive finite element space directly, we only need to do some smoothing steps for a boundary value problems on each adaptive space and solve some eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the eigenvalue problems on each adaptive space which is time-consuming. Further, the complementary error estimate for eigenvalue problem will be introduced. This estimate can not only provide an accurate error estimate for eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.

Fei Xu, Qiumei Huang, Shuangshuang Chen & Hongkun Ma. (2020). A Type of Cascadic Adaptive Finite Element Method for Eigenvalue Problem. Advances in Applied Mathematics and Mechanics. 12 (3). 774-796. doi:10.4208/aamm.OA-2019-0054
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