JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (08): 34-39.doi: 10.6040/j.issn.1671-9352.0.2015.055

Previous Articles     Next Articles

Local multigrid method for higher-order finite element discretizations of elasticity problems in two dimensions

LIU Chun-mei1, ZHONG Liu-qiang2, SHU Shi3, XIAO Ying-xiong4   

  1. 1. Institute of Computational Mathematics, College of Science, Hunan University of Science and Engineering, Yongzhou 425199, Hunan, China;
    2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China;
    3. School of Mathematical and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China;
    4. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, Hunan, China
  • Received:2015-02-02 Online:2015-08-20 Published:2015-07-31

PDF (PC)

525

Abstract: Due to few limited vertexes increase during every refinement of adaptive finite element method (AFEM), only some limited basis functions change between two finite element spaces based on two adjacent refinement meshes. By use of this special property, a type of multigrid method based on the local relaxation is applied to the high-order AFEM discrete systems of elasticity problems in two dimensions, that is, during each iteration, the part of the homogeneous high-order systems based on hierarchical basis functions is solved by a symmetric Gauss-Seidal method once, and then these residual systems are projected onto linear finite element space, and discrete systems based on linear finite element spaces are generated. Finally, these linear finite element discretizations are solved by a symmetric local Gauss-Seidal method once. The numerical experiments show that the local multigrid method is robust.

Key words: elasticity problems, higher-order finite element discretizations, local relaxation, multigrid method

CLC Number: 

  • O241.8
[1] SENTURIA S, ALURU N, WHITE J. Simulating the behavior of MEMS devices: computational methods and needs[J]. IEEE Computational Science and Engineering, 1997, 4(1):30-43.
[2] SENTURIA S, HARRIS R, JOHNSON B, et al. A computer-aided design system for microelectromechanical systems[J]. Journal of Microelectro Mechanical Systems, 1992, 1(1):3-13.
[3] BRENNER S C, SUNG Liyeng. Linear finite element methods for planar linear elasticity[J]. Mathematics of Computation, 1992, 59:321-338.
[4] 陈竹昌,王建华,王卫中. 自适应多层网格有限元求解应力集中问题[J].同济大学学报:自然科学版,1994,22(3):203-208. HEN Zhuchang, WANG Jianhua, WANG Weizhong. Adaptive multigrid FEM for stress concentration[J].Journal of Tongji University:Natural Sciences, 1994, 22(3):203-208.
[5] 梁力,林韵梅. 有限元网格修正的自适应分析及其应用[J]. 工程力学, 1995,12(2):109-118. LIANG Li, LIN Yunmei. Adaptive mesh refinement of finite element method and its application[J]. Engineering Mechanics, 1995, 12(2):109-118.
[6] 王建华. 线弹性有限元的自适应加密与多层网格法求解[J]. 河海大学学报:自然科学版,1994,22(3):16-22. WANG Jianhua. Adaptive refinement and multigrid solution for linear finite element method[J]. Journal of Hohai University:Natural Sciences, 1994, 22(3):16-22.
[7] 王建华, 殷宗泽, 赵维炳. 自适应多层网格有限元网格生成器研制[J]. 计算结构力学与其应用, 1995, 12(1):86-92. WANG Jianhua, YIN Zongze, ZHAO Weibing. Implementation of the mesh generator for adaptive multigrid finite element method[J]. Computational Structural Mechanics and Applications, 1995, 12(1):86-92.
[8] CAI Zhiqiang, KORSAWE J, STARKE G. An adaptive least squares mixed finite element method for the stress displacement formulation of linear elasticity[J]. Numerical Methods Partial Differential Equations, 2005, 21(1):132-148.
[9] WHILER T P. Locking-Free Adaptive discontinuous Galerkin FEM for linear elasticity problem[J]. Mathematics of Computation, 2006, 75(255):1087-1102.
[10] CHEN Long, ZHANG Chensong. A coarsening algorithm on adaptive grids by newest vertex bisection and its applications[J]. Journal of Computational Mathematics, 2010, 28(6):767-789.
[11] 刘春梅,肖映雄,舒适,等. 弹性力学问题自适应有限元及其局部多重网格法[J]. 工程力学,2012,29(9):60-67. LIU Chunmei, XIAO Yingxiong, SHU Shi, et al. Adaptive finite element method and local multigrid method for elasticity problem[J]. Engineering Mechanics, 2012, 29(9):60-67.
[1] DING Feng-xia, CHENG Hao. A posteriori choice rule for the mollification regularization parameter for the Cauchy problem of an elliptic equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 18-24.
[2] YU Jin-biao, REN Yong-qiang, CAO Wei-dong, LU Tong-chao, CHENG Ai-jie, DAI tao. Expanded mixed finite element method for compressible miscible displacement in heterogeneous porous media [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 25-34.
[3] WANG Yang, ZHAO Yan-jun, FENG Yi-fu. On successive-overrelaxation acceleration of MHSS iterations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(8): 61-65.
[4] NIE Tian-yang, SHI Jing-tao. The connection between DPP and MP for the fully coupled forward-backward stochastic control systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(5): 121-129.
[5] LIU Wen-yue, SUN Tong-jun. Iterative non-overlapping domain decomposition method for optimal boundary control problems governed by elliptic equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 21-28.
[6] CHEN Yi-ming, KE Xiao-hong, HAN Xiao-ning, SUN Yan-nan, LIU Li-qing. Wavelets method for solving system of fractional differential equations and the convergence analysis [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(02): 67-74.
[7] SU Li-Juan, Wang-Wen-He. A finite difference method for the two sided space fractional advection diffusion equation [J]. J4, 2009, 44(10): 26-29.
[8] LI Zhi-Chao, FU Gong-Fei. The dynamic finite element method with characteristics for convectiondominated diffusion problems [J]. J4, 2009, 44(8): 90-96.
[9] ZHANG Xin-Dong, HU Yu-Hong. On the exact solution of a diffusion equation with boundary conditions by the homotopy analysis method [J]. J4, 2008, 43(12): 73-76.
[10] ZUO Jin-ming . A computational method for the nonlinear BBM equation [J]. J4, 2008, 43(4): 47-50 .
[11] ZUO Jin-ming . Computational method for the generalized KdV equation [J]. J4, 2008, 43(6): 44-48 .
[12] XU Qiu-yan .

A class of parallel finite difference methods for solving a two-dimensional diffusion equation

[J]. J4, 2008, 43(8): 1-05 .
[13] ZHANG Qing-jie,WANG Wen-qia . A high order parallel iterative scheme for the dispersive equation [J]. J4, 2008, 43(2): 8-11 .
[14] SUN Peng and ZHAO Wei-dong . Alternatingdirection implicit upwind finite volume method for pricing Asian options [J]. J4, 2007, 42(6): 16-21 .
[15] CHANG Yan-zhen and YANG Dan-ping . Finite element analysis of a coupled thermally dependent viscosity Stokes flow problem [J]. J4, 2007, 42(8): 9-16 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd