体积约束的非局部扩散问题的后验误差分析

doi:10.12286/jssx.2018.1.107

计算数学 ›› 2018, Vol. 40 ›› Issue (1): 107-116.DOI: 10.12286/jssx.2018.1.107

• 论文 • 上一篇

体积约束的非局部扩散问题的后验误差分析

葛志昊, 吴慧丽

河南大学数学与统计学院 & 应用数学所, 开封 475004

收稿日期:2017-04-11 出版日期:2018-03-15 发布日期:2018-02-03
基金资助:
河南省自然科学基金（No：162300410031），河南大学优秀青年资助项目（No：yqpy20140039）.

PDF ( 556 ) 引用被引次数 ( 1 | 1 )

葛志昊, 吴慧丽. 体积约束的非局部扩散问题的后验误差分析[J]. 计算数学, 2018, 40(1): 107-116.

Ge Zhihao, Wu Huili. A POSTERIORI ERROR ANALYSIS OF NONLOCAL DIFFUSION PROBLEM WITH VOLUME CONSTRAINTS[J]. Mathematica Numerica Sinica, 2018, 40(1): 107-116.

A POSTERIORI ERROR ANALYSIS OF NONLOCAL DIFFUSION PROBLEM WITH VOLUME CONSTRAINTS

Ge Zhihao, Wu Huili

School of Mathematics and Statistics & Institute of Applied Mathematics, Henan University, Kaifeng 475004, China

Received:2017-04-11 Online:2018-03-15 Published:2018-02-03

本文针对体积约束的非局部扩散问题构造了新的后验误差指示器，证明了后验误差指示器的可靠性以及有效性.数值算例验证了理论结果.

关键词： 非局部扩散问题, 有限元方法, 后验误差估计

In this paper, we propose a new posteriori error estimator for the nonlocal diffusion problem with volume constraints. And we prove the reliability and efficiency of the posteriori error estimator. Finally, we give the numerical experiments to verify the theoretical results.

Key words: Nonlocal diffusion problem, finite element method, posteriori error estimate.

MR(2010)主题分类:

65R20
45L05

分享此文:

[1] Kroner E. Elasticity theory of materials with long range forces[J]. Int. J. Soli. Struc., 1967, 3(5):731-742.

[2] Rogula D. Nonlocal theory of material media[M]. Springer, Berlin, 1982.

[3] Altan B. Uniqueness of initial-boundary value problems in nonlocal elasticity[J]. Int. J. Soli. Struc., 1989, 25(11):1271-1278.

[4] Altan B. Uniqueness in nonlocal thermoelasticity[J]. J. Ther. Stre., 1991, 14(2):121-128.

[5] Silling S. Reformulation of elasticity theory for discontinuities and long-range forces[J]. J. Mech. Phys. Soli., 2000, 48(1):175-209.

[6] Silling S. Dynamic frature modeling with a meshfree peridynamic code[J]. Comput. Fluid and Solid Mech., 2003, 2003:641-644.

[7] Gilboa G., Osher S. Nonlocal operators with applications to image processing[J]. SIAM J. Mult. Model. Simu., 2008, 7(3):1005-1028.

[8] Du Q., Zhou K. Mathematical analysis for the peridynamics nonlocal continuum theory[J]. ESAIM Math. Model. Numer. Anal., 2011, 45(2):217-234.

[9] Emmrich E., Weckner O. On the well-posedness of the linear peridynamic model and its convergence towards the navier equation of linear elasticity[J]. Comm. Math. Sci., 2007, 5(4):851-864.

[10] Emmrich E., Weckner O. Analysis and numerical approximation of an integro-differential equation modeling nonlocal effects in linear elasticity[J]. Math. and Mech. Solids, 2007, 4(4):363-384.

[11] Lehoucq R., Silling S. Force flux and the peridynamic stress tensor[J]. J. Mech. and Phys. Solids, 2008, 56(4):1566-1577.

[12] Gunzburger M., Lehoucq R. A nonlocal vector calculus with application to nonlocal boundary value problems[J]. SIAM J. Mult. Model. Simu., 2010, 8(5):1581-1598.

[13] Du Q., Gunzbueger M., Lehoucq R., Zhou K. A nonlocal vector caculus, nonlocal volumeconstrained problems, and the nonlocal balance laws[J]. Math. Models Meth. Appl. Sci., 2013, 23(3):493-540.

[14] Aksoylu B., Parks M. Variational theory and domain decomposition for nonlocal problems[J]. Appl. Math. and Comp., 2011, 217(14):6498-6515.

[15] Bobaru F., Yang M., Alves L., Silling S., Askari E., Xu J. Convergence, adaptive refinement, and scaling in 1d peridynamics[J]. Int. J. Numer. Meth. Eng., 2009, 77(6):852-877.

[16] Chen X., Gunzburger M. Continuous and discontinuous finite element methods for a peridynamics model of mechanics[J]. Comp. Meth. Appl. Mech. Eng., 2011, 200(9-12):1237-1250.

[17] Macek R., Silling S. Peridynamics via finite element analysis[J]. Fini. Elem. Anal. Desi., 43(15):1169-1178.

[18] Silling S., Askari E. A meshfree method based on the peridynamic model of solid mechanics[J]. Comp. and Stru., 2005, 83(17-18):1526-1535.

[19] Du Q., Tian L., Zhao X. A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models[J]. SIAM J. Numer. Anal., 2013, 51(2):1211-1234.

[20] Du Q., Ju L., Tian L., Zhou K. A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models[J]. Math. Comp., 2013, 82(284):1889-1992.

[1]	高兴华, 李宏, 刘洋. 分布阶扩散—波动方程的有限元解的误差估计[J]. 计算数学, 2021, 43(4): 493-505.
[2]	董自明, 李宏, 赵智慧, 唐斯琴. 对流扩散反应方程的局部投影稳定化连续时空有限元方法[J]. 计算数学, 2021, 43(3): 367-387.
[3]	曾玉平, 翁智峰, 胡汉章. 简化摩擦接触问题的对称弱超内罚间断Galerkin方法的先验和后验误差估计[J]. 计算数学, 2021, 43(2): 162-176.
[4]	房明娟, 阳莺, 唐鸣. 稳态Poisson-Nernst-Planck方程的残量型后验误差估计[J]. 计算数学, 2021, 43(1): 17-32.
[5]	唐斯琴, 李宏, 董自明, 赵智慧. 对流反应扩散方程的稳定化时间间断时空有限元解的误差估计[J]. 计算数学, 2020, 42(4): 472-486.
[6]	洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309.
[7]	戴小英. 电子结构计算的数值方法与理论[J]. 计算数学, 2020, 42(2): 131-158.
[8]	何斯日古楞, 李宏, 刘洋, 方志朝. 非稳态奇异系数微分方程的时间间断时空有限元方法[J]. 计算数学, 2020, 42(1): 101-116.
[9]	张然. 弱有限元方法在线弹性问题中的应用[J]. 计算数学, 2020, 42(1): 1-17.
[10]	李世顺, 祁粉粉, 邵新平. 求解定常不可压Stokes方程的两层罚函数方法[J]. 计算数学, 2019, 41(3): 259-265.
[11]	王俊俊, 李庆富, 石东洋. 非线性抛物方程混合有限元方法的高精度分析[J]. 计算数学, 2019, 41(2): 191-211.
[12]	武海军. 高波数Helmholtz方程的有限元方法和连续内罚有限元方法[J]. 计算数学, 2018, 40(2): 191-213.
[13]	李宏, 杜春瑶, 赵智慧. 反应扩散方程的连续时空有限元方法[J]. 计算数学, 2017, 39(2): 167-178.
[14]	李晓翠, 杨小远, 张英晗. 一类随机非自伴波方程的半离散有限元近似[J]. 计算数学, 2017, 39(1): 42-58.
[15]	曹济伟. 求解二维时谐Maxwell方程的一种混合有限元新格式[J]. 计算数学, 2016, 38(4): 429-441.

阅读次数

全文

摘要

体积约束的非局部扩散问题的后验误差分析

A POSTERIORI ERROR ANALYSIS OF NONLOCAL DIFFUSION PROBLEM WITH VOLUME CONSTRAINTS

扫码分享