Extended Fisher-Kolmogorov方程的一类低阶非协调混合有限元方法
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摘要
该文的主要目的是研究Extended Fisher-Kolmogorov(EFK)方程的一类低阶非协调元混合有限元方法.首先引入一个中间变量v=-△u将原方程分裂为两个二阶方程,建立了一个非协调混合元逼近格式,并通过构造一个李雅普诺夫泛函证明了半离散格式逼近解的一个先验估计并证明了解的存在唯一性.在半离散格式下,利用这个先验估计和单元的性质,证明了原始变量u和中间变量v的H~1-模意义下的最优误差估计.进一步地,借助高精度技巧得到了O(h~2)阶的超逼近性质.其次,建立了一个新的线性化的向后Euler全离散格式,通过对相容误差和非线性项采用新的分裂技术,导出了u和v的H~1-模意义下具有O(h+τ)和O(h~2+τ)的最优误差估计和超逼近结果.这里,h,τ分别表示空间剖分参数和时间步长.最后,给出了一个数值算例,计算结果验证了理论分析的正确性,该文的分析为利用非协调混合有限元研究其它四阶初边值问题提供了一个可借鉴的途径.
The purpose of this paper is to study the mixed finite element methods with a type of lower order nonconforming finite elements for the extended Fisher-Kolmogorov(EFK)equation. Firstly, a nonconforming mixed finite element scheme is established by splitting the EFK equation into two second order equations through a intermediate variable v =-△u. Some a priori bounds are derived by use of Lyapunov functional, existence and uniqueness for the approximation solutions are also proved. The optimal error estimates for both the primitive solution u and the intermediate variable v in H~1-norm are deduced for semi-discrete scheme by use of the above priori bounds and properties of the elements. Furthermore, the superclose properties with order O(h~2) are obtained through high accuracy results of the elements. Secondly, a new linearized backward Euler full-discrete scheme is established. The optimal error estimates and superclose results for u and v in H~1-norm with orders O(h+τ) and O(h~2 +τ) are obtained respectively through the new splitting techniques for consistency errors and nonlinear terms. Here, h, τ are parameter of the subdivision in space and time step. Finally, numerical results are provided to confirm the theoretical analysis. Our analysis provides a new understanding with nonconforming mixed finite element methods to analyze other fourth order initial boundary value problems.
The purpose of this paper is to study the mixed finite element methods with a type of lower order nonconforming finite elements for the extended Fisher-Kolmogorov(EFK)equation. Firstly, a nonconforming mixed finite element scheme is established by splitting the EFK equation into two second order equations through a intermediate variable v =-△u. Some a priori bounds are derived by use of Lyapunov functional, existence and uniqueness for the approximation solutions are also proved. The optimal error estimates for both the primitive solution u and the intermediate variable v in H~1-norm are deduced for semi-discrete scheme by use of the above priori bounds and properties of the elements. Furthermore, the superclose properties with order O(h~2) are obtained through high accuracy results of the elements. Secondly, a new linearized backward Euler full-discrete scheme is established. The optimal error estimates and superclose results for u and v in H~1-norm with orders O(h+τ) and O(h~2 +τ) are obtained respectively through the new splitting techniques for consistency errors and nonlinear terms. Here, h, τ are parameter of the subdivision in space and time step. Finally, numerical results are provided to confirm the theoretical analysis. Our analysis provides a new understanding with nonconforming mixed finite element methods to analyze other fourth order initial boundary value problems.
引文
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[2]Dee G T,Van S W.Bistable systems with propagating fronts leading to pattern formation.Phys Rev Lett,1988,60(25):2641-2644
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[4]Hornreich R M,Luban M,Shtrikman S.Critical behaviour at the onset of k-space instability on theλline.Phys Rev Lett,1975,35(22):1678-1681
[5]Kadri T,Omrani K.A second-order accurate difference scheme for an extended Fisher-Kolmogorov equation.Comput Math Appl,2011,61(2):451-459
[6]Danumjaya P,Pani A K.Numerical methods for the Existended Fisher-Kolmogorov(EFK)equation.Int J Numer Anal Model,2006,3(2):186-210
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[10]石东洋,史艳华,王芬玲.四阶抛物方程的H~1-Galerkin混合有限元方法的超遍近及最优误差估计.计算数学,2014,36(4):363-380Shi D Y,Shi Y H,Wang F L.Supercloseness and the optimal order error estimates of H~1-Galerkin mixed finite element method for fourth order parabolic equation.Math Numer Sini,2014,36(4):363-380
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[13]陈绍春,陈红如.二阶椭圆问题新的混合元格式.计算数学,2010,32(2):213-218Chen S C,Chen H R.New mixed element schemes for second order elliptic problem.Math Numer Sini,2010,32(2):213-218
[14]史峰,于佳平,李开泰.椭圆型方程的一种新型混合有限元格式.工程数学学报,2011,28(2):231-237Shi F,Yu J P,Li K T.A new mixed finite element scheme for elliptic equation.J Enging Math,2011,28(2):231-237
[15]石东洋,李明浩.二阶椭圆问题一种新格式的高精度分析.应用数学学报,2014,37(1):45-58Shi D Y,Li M H.High accuracy analysis of new schemes for second order elliptic problem for recurrent event data.Acta Math Appl Sini,2014,37(1):45-58
[16]Wang J F,Li H,He Siriguleng.A New linearized Crank-Nicolson mixed element scheme for the Extended Fisher-Kolmogorov equation.The Scientific World Journal,2013,2013(1):202-212
[17]Chen Z X.Expanded mixed finite element methods for linear second order elliptic problems I.ESAIM Math Model Numer Anal,1998,32(4):479-499
[18]Danumjaya P,Pani A K.Mixed finite element methods for a fourth order reaction diffusion equation.Numer Methods Partial Differential Equations,2012,28(4):1227-1251
[19]Lin Q,Tobiska L,Zhou A H.Superconvergence and extrapolation of nonconformimg low order finite elements applied to the Poisson equation.IMA J Numer Anal,2005,25(1):160-181
[20]Shi D Y.Mao S P.Chen S C.An anisotropic nonconforming finite element with some superconvergence results.J Comput Math,2005,23(3):261-274
[21]Shi D Y,Ren J C.Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes.Nonlinear Anal,2009,71(9):3842-3852
[22]Shi D Y,Zhang B Y.High accuracy analysis of the finite element method for nonlinear viscoelastic wave equations with nonlinear boundary conditions.J Syst Sci Complex,2011,24(4):795-802
[23]张铁.Cahn-Hilliard方程的有限元分析.计算数学,2006,28(3):281-292Zhang T.Finite element analysis for Cahn-Hilliard equation.Math Numer Sini,2006,28(3):281-292
[24]石东洋,张厚超,王瑜.一类非线性四阶双曲方程扩展的混合元方法的超收敛分析.计算数学,2016,38(1):65-82Shi D Y,Zhang H C,Wang Y.Superconvergence analysis of an expanded mixed finite element method for nonlinear fourth-order hyperbolic equation.Math Numer Sini,2016,38(1):65-82
[25]张厚超,石东洋.非线性四阶双曲方程低阶混合元方法的超收敛分析.数学物理学报,2016,36A(4):656-671Zhang H C,Shi D Y.Superconvergence analysis of a lower order mixed finite element method for nonlinear fourth-order hyperbolic equation.Acta Math Sci,2016,36A(4):656-671
[26]Rannacher R,Turek S.Simple nonconforming quadrilateral Stokes element.Numer Methods Partial Differential Equations,1992,8(2):97-111
[27]Park C J,Sheen D W.P_1-nonconforming quadrilateral finite element method for second order elliptic problems.SIAM J Numer Anal,2003,41(2):624-640
[28]Hu J,Shi Z C.Constrained quadrilateral nonconforming rotated Q_1 element.J Comput Math,2005,23(5):561-586
[29]Lin Q,Lin J F.Finite Element Method:Accuracy and Improvement.Beijing:Scicence Press,2006
[30]Shi D Y,Xu C.Anisotropic nonconforming Crouzeix-Raviart type FEM for second order elliptic problems.Appl Math Mech Eng-Ed,2012,33(2):243-252
[31]Shi D Y,Wang F L,Zhao Y M.Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations.Acta Math Appl Sin Engl Ser,2013,29(2):403-414
[32]Shi D Y,Hao X B.Accuracy analysis for quasi-Carey element.J Syst Sci Complex,2008,21(3):456-462
[33]Shi Z C,Jiang B,Xue W M.A new superconvergence property of Wilson nonconforming finite element.Numer Math,1997,78(2):259-268
[34]Shi D Y,Chen S C,I.Hagiwara.Convergence analysis for a nonconforming membrane element on anistropic meshes.J Comput Math,2005,23(4):373-382
[35]Khiari N,Omrani K.Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions.Comput Math Appl,2011,62(11):4151-4160
[2]Dee G T,Van S W.Bistable systems with propagating fronts leading to pattern formation.Phys Rev Lett,1988,60(25):2641-2644
[3]Aronson D G,Weinberger H F.Multidimensional nonlinear diffusion arising in population genetics.Adv Math,1978,30(1):33-76
[4]Hornreich R M,Luban M,Shtrikman S.Critical behaviour at the onset of k-space instability on theλline.Phys Rev Lett,1975,35(22):1678-1681
[5]Kadri T,Omrani K.A second-order accurate difference scheme for an extended Fisher-Kolmogorov equation.Comput Math Appl,2011,61(2):451-459
[6]Danumjaya P,Pani A K.Numerical methods for the Existended Fisher-Kolmogorov(EFK)equation.Int J Numer Anal Model,2006,3(2):186-210
[7]Pei L F.Research on new C~0-nonconforming finite element schemes and superconvergence analysis[D].2014:66-82
[8]Li J.Optimal convergence analysis of mixed finite element methods for a fourth-order elliptic and parabolic problems.Numer Methods Partial Differential Equations,2006,22(4):884-896
[9]李宏,刘洋.一类四阶抛物型积分-微分方程的混合间断时空有限元方法.计算数学,2007,29(4):413-420Li H,Liu Y.Mixed discontionous space-time finite element method for the fourth order parabolic integrodifferential equations.Math Numer Sini,2007,29(4):413-420
[10]石东洋,史艳华,王芬玲.四阶抛物方程的H~1-Galerkin混合有限元方法的超遍近及最优误差估计.计算数学,2014,36(4):363-380Shi D Y,Shi Y H,Wang F L.Supercloseness and the optimal order error estimates of H~1-Galerkin mixed finite element method for fourth order parabolic equation.Math Numer Sini,2014,36(4):363-380
[11]刘洋,李宏,何斯日古楞,等,四阶抛物偏微分方程的H~1-Galerkin混合元方法及数值模拟.计算数学,2012,34(4):259-274Liu Y,Li H,He S,et al.H~1-Galerkin mixed element method and numerical simulation for the fourth order parabolic partial differential equations.Math Numer Sini,2012,34(4):259-274
[12]何斯日古楞,李宏.带广义边界条件的四阶抛物型方程的混合间断时空有限元方法.计算数学,2009,31(2):167-178He Siriguleng,Li H.The mixed discontinuous space-time finite element method for the fourth order linear parabolic equation with generalized boundary condition.Math Numer Sini,2009,31(2):167-178
[13]陈绍春,陈红如.二阶椭圆问题新的混合元格式.计算数学,2010,32(2):213-218Chen S C,Chen H R.New mixed element schemes for second order elliptic problem.Math Numer Sini,2010,32(2):213-218
[14]史峰,于佳平,李开泰.椭圆型方程的一种新型混合有限元格式.工程数学学报,2011,28(2):231-237Shi F,Yu J P,Li K T.A new mixed finite element scheme for elliptic equation.J Enging Math,2011,28(2):231-237
[15]石东洋,李明浩.二阶椭圆问题一种新格式的高精度分析.应用数学学报,2014,37(1):45-58Shi D Y,Li M H.High accuracy analysis of new schemes for second order elliptic problem for recurrent event data.Acta Math Appl Sini,2014,37(1):45-58
[16]Wang J F,Li H,He Siriguleng.A New linearized Crank-Nicolson mixed element scheme for the Extended Fisher-Kolmogorov equation.The Scientific World Journal,2013,2013(1):202-212
[17]Chen Z X.Expanded mixed finite element methods for linear second order elliptic problems I.ESAIM Math Model Numer Anal,1998,32(4):479-499
[18]Danumjaya P,Pani A K.Mixed finite element methods for a fourth order reaction diffusion equation.Numer Methods Partial Differential Equations,2012,28(4):1227-1251
[19]Lin Q,Tobiska L,Zhou A H.Superconvergence and extrapolation of nonconformimg low order finite elements applied to the Poisson equation.IMA J Numer Anal,2005,25(1):160-181
[20]Shi D Y.Mao S P.Chen S C.An anisotropic nonconforming finite element with some superconvergence results.J Comput Math,2005,23(3):261-274
[21]Shi D Y,Ren J C.Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes.Nonlinear Anal,2009,71(9):3842-3852
[22]Shi D Y,Zhang B Y.High accuracy analysis of the finite element method for nonlinear viscoelastic wave equations with nonlinear boundary conditions.J Syst Sci Complex,2011,24(4):795-802
[23]张铁.Cahn-Hilliard方程的有限元分析.计算数学,2006,28(3):281-292Zhang T.Finite element analysis for Cahn-Hilliard equation.Math Numer Sini,2006,28(3):281-292
[24]石东洋,张厚超,王瑜.一类非线性四阶双曲方程扩展的混合元方法的超收敛分析.计算数学,2016,38(1):65-82Shi D Y,Zhang H C,Wang Y.Superconvergence analysis of an expanded mixed finite element method for nonlinear fourth-order hyperbolic equation.Math Numer Sini,2016,38(1):65-82
[25]张厚超,石东洋.非线性四阶双曲方程低阶混合元方法的超收敛分析.数学物理学报,2016,36A(4):656-671Zhang H C,Shi D Y.Superconvergence analysis of a lower order mixed finite element method for nonlinear fourth-order hyperbolic equation.Acta Math Sci,2016,36A(4):656-671
[26]Rannacher R,Turek S.Simple nonconforming quadrilateral Stokes element.Numer Methods Partial Differential Equations,1992,8(2):97-111
[27]Park C J,Sheen D W.P_1-nonconforming quadrilateral finite element method for second order elliptic problems.SIAM J Numer Anal,2003,41(2):624-640
[28]Hu J,Shi Z C.Constrained quadrilateral nonconforming rotated Q_1 element.J Comput Math,2005,23(5):561-586
[29]Lin Q,Lin J F.Finite Element Method:Accuracy and Improvement.Beijing:Scicence Press,2006
[30]Shi D Y,Xu C.Anisotropic nonconforming Crouzeix-Raviart type FEM for second order elliptic problems.Appl Math Mech Eng-Ed,2012,33(2):243-252
[31]Shi D Y,Wang F L,Zhao Y M.Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations.Acta Math Appl Sin Engl Ser,2013,29(2):403-414
[32]Shi D Y,Hao X B.Accuracy analysis for quasi-Carey element.J Syst Sci Complex,2008,21(3):456-462
[33]Shi Z C,Jiang B,Xue W M.A new superconvergence property of Wilson nonconforming finite element.Numer Math,1997,78(2):259-268
[34]Shi D Y,Chen S C,I.Hagiwara.Convergence analysis for a nonconforming membrane element on anistropic meshes.J Comput Math,2005,23(4):373-382
[35]Khiari N,Omrani K.Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions.Comput Math Appl,2011,62(11):4151-4160