非线性Poisson-Boltzmann方程的改进无单元Galerkin法分析
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摘要
【目的】利用改进无单元Galerkin法求解非线性Poisson-Boltzmann方程。【方法】将改进的移动最小二乘近似与非线性Poisson-Boltzmann方程的Galerkin弱形式耦合,建立了非线性Poisson-Boltzmann方程的改进无单元Galerkin法。基于改进移动最小二乘近似的误差结果下,推导了非线性Poisson-Boltzmann方程的改进无单元Galerkin法的误差估计。【结果】在Sobolev空间中获得了误差估计,并通过数值算例验证了理论结果。【结论】该方法具有较高的计算精度和较好的稳定性,误差随节点间距的减小而降低。
[Purposes]The nonlinear Poisson-Boltzmann equation is solved and analyzed by the improved element-free Galerkin method.[Methods]Combining the improved moving least square approximation with Galerkin weak form,the improved element-free Galerkin method is establish for the nonlinear Poisson-Boltzmann equation.Based on the error results of the improved moving least square approximation,the error of the improved element-free Galerkin method for nonlinear Poisson-Boltzmann equation is derived theoretically.[Findings]Error estimation is obtained in the Sobolev space.Numerical examples verify the theoretical analysis.[Conclusions]The method has higher calculation accuracy and better stability.The errors decrease as the nodal spacing reduces.
[Purposes]The nonlinear Poisson-Boltzmann equation is solved and analyzed by the improved element-free Galerkin method.[Methods]Combining the improved moving least square approximation with Galerkin weak form,the improved element-free Galerkin method is establish for the nonlinear Poisson-Boltzmann equation.Based on the error results of the improved moving least square approximation,the error of the improved element-free Galerkin method for nonlinear Poisson-Boltzmann equation is derived theoretically.[Findings]Error estimation is obtained in the Sobolev space.Numerical examples verify the theoretical analysis.[Conclusions]The method has higher calculation accuracy and better stability.The errors decrease as the nodal spacing reduces.
引文
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[11]LI X L,WANG Q Q.Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases[J].Engineering Analysis with Boundary Elements,2016,73:21-34.
[12]LI X L,ZHANG S.Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving leastsquare approximation[J].Applied Mathematical Modelling,2016,40(4):2875-2896.
[13]ZHANG Z,WANG J F,CHENG Y M,et al.The improved element-free Galerkin method for three-dimensional transient heat conduction problems[J].Science China(Physics Mechanics&Astronomy),2013,56(8):1568-1580.
[14]ZHANG L W,DENG Y J,LIEW K M.An improved element-free Galerkin method for numerical modeling of the biological population problems[J].Engineering Analysis with Boundary Elements,2014,40(40):181-188.
[15]TANG Y Z,LI X L.Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems[J].Chinese Physics B,2017,26(3):215-225.
[16]LI X L,CHEN H,WANG Y.Error analysis in sobolev spaces for the improved moving least-squre approximation and the improved element-free Galerkin method[J].Applied Mathematics and Computation,2015,262:56-78.
[17]LI X L.Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces[J].Applied Numerical Mathematics,2016,99:77-97.
[18]ZHU T,ATLURI S N.A modified collocation method and apenalty formulation for enforcing the essential boundary conditions in the element free Galerkin method[J].Computational Mechanics,1998,21(3):211-222.
[2]LU B Z,CHENG X L,HUANG J F.AFMPB:An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems[J].Computer Physics Communications,2010,181(6):1150-1160.
[3]BOSCHITSCH A H,FENLEY M O.Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation[J].Journal of Computational Chemistry,2004,25(7):935-955.
[4]QIAO Z H,LI Z L,TANG T.A finite difference scheme for solving the nonlinear Poisson-Boltzmann equation modeling charged spheres[J].Comput Math,2006,24(3):252-264.
[5]CHEN L,HOST M J,XU J C.The finite element approximation of the nonlinear Poisson-Boltzmann equation[J].SI-AM Journal on Numerical Analysis,2007,45(6):2298-2320.
[6]HOLST M,Mc CAMMOM J A,YU Z,et al..Adaptive finite element modeling techniques for the Poisson-Boltzmann equation[J].Communications in Computational Physics.2012,11(1):179-214.
[7]程玉民.无网格方法[M].北京:科学出版社,2015.CHENG Y M.Meshless method[M].Beijing:Science Press,2015.
[8]NAYROLES B,TOUZOT G,VILLON P.Generalizing the finite element method:diffuse approximation and diffuse elements[J].Computational Mechanics,1992,10(5):307-318.
[9]BELYTSCHKO T,LU Y Y,GU L.Element-free Galerkin methods[J].International Journal for Numerical Methods in Engineering,1994,37(2):229-256.
[10]LI X L,LI S L.On the stability of the moving least squares approximation and element-free Galerkin method[J].Computers and Mathematics with Applications,2016,72(6):1515-1531.
[11]LI X L,WANG Q Q.Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases[J].Engineering Analysis with Boundary Elements,2016,73:21-34.
[12]LI X L,ZHANG S.Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving leastsquare approximation[J].Applied Mathematical Modelling,2016,40(4):2875-2896.
[13]ZHANG Z,WANG J F,CHENG Y M,et al.The improved element-free Galerkin method for three-dimensional transient heat conduction problems[J].Science China(Physics Mechanics&Astronomy),2013,56(8):1568-1580.
[14]ZHANG L W,DENG Y J,LIEW K M.An improved element-free Galerkin method for numerical modeling of the biological population problems[J].Engineering Analysis with Boundary Elements,2014,40(40):181-188.
[15]TANG Y Z,LI X L.Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems[J].Chinese Physics B,2017,26(3):215-225.
[16]LI X L,CHEN H,WANG Y.Error analysis in sobolev spaces for the improved moving least-squre approximation and the improved element-free Galerkin method[J].Applied Mathematics and Computation,2015,262:56-78.
[17]LI X L.Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces[J].Applied Numerical Mathematics,2016,99:77-97.
[18]ZHU T,ATLURI S N.A modified collocation method and apenalty formulation for enforcing the essential boundary conditions in the element free Galerkin method[J].Computational Mechanics,1998,21(3):211-222.