体积约束的非局部扩散问题的后验误差分析
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摘要
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.
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.
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.
引文
[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 applicat.ions 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.
[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 applicat.ions 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.