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）.

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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

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[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.

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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

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