非线性强阻尼波动方程一个新的H~1-Galerkin混合有限元分析
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摘要
利用不完全双二次元Q_2~-和一阶BDFM元,对一类非线性强阻尼波动方程建立了一个新的混合元逼近模式.借助这两个单元的插值算子的特殊性质和平均值技巧,对半离散和线性化Euler全离散格式,分别导出了原始变量在H~1-模和中间变量在H(div)-模意义下具有O(h~3)和O(h~3+τ~2)阶的超逼近估计,比以往文献的最优误差估计高一阶.
In this paper,H~1-Galerkin mixed finite element method for a kind of nonlinear strongly damped wave equations was studied.A new mixed finite element pattern was developed with incomplete biquadratic element Q_2~- and first order BDFM element.With the help of the special properties of the interpolation operators of these two elements and mean-value technique,the superclose estimates for the primitive variable in H~1-norm and the intermediate variable in H(div)-norm were deduced respectively for the semi-discrete and the linearized fully discrete schemes,which were one order higher than the corresponding optimal error estimations in the existing literature published before.
In this paper,H~1-Galerkin mixed finite element method for a kind of nonlinear strongly damped wave equations was studied.A new mixed finite element pattern was developed with incomplete biquadratic element Q_2~- and first order BDFM element.With the help of the special properties of the interpolation operators of these two elements and mean-value technique,the superclose estimates for the primitive variable in H~1-norm and the intermediate variable in H(div)-norm were deduced respectively for the semi-discrete and the linearized fully discrete schemes,which were one order higher than the corresponding optimal error estimations in the existing literature published before.
引文
[1]尚亚东.方程utt-Δutt-Δut-Δu=f(u)的初边值问题[J].应用数学学报,2000,23(3):385-392.
[2]刘洋,李宏.四阶强阻尼波动方程的新的混合有限元方法[J].计算数学,2010,32(2):157-170.
[3]方志朝,刘洋,李宏.四阶强阻尼波动方程的混合控制体积法[J].计算数学,2011,33(4):409-422.
[4]张亚东,李新祥,石东洋.强阻尼波动方程的非协调有限元超收敛分析[J].山东大学学报(理学版),2014,49(5):28-35.
[5]罗振东.混合有限元方法基础及其应用[M].北京:科学出版社,2006.
[6]陈绍春,陈红如.二阶椭圆问题新的混合元格式[J].计算数学,2010,32(2):213-218.
[7]石东洋,李明浩.二阶问题一种新格式的高精度分析[J].应用数学学报,2014,37(1):45-58.
[8]石东洋,张亚东.抛物型方程一个新的非协调混合元超收敛性分析及外推[J].计算数学,2013,35(4):337-352.
[9]Shi D Y,Zhang Y D.High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equation[J].Appl Math Comput,2011,218(7):3176-3186.
[10]史艳华,石东洋.Sobolev方程新混合元方法的高精度分析[J].系统科学与数学,2014,34(4):452-463.
[11]毛凤梅,刁群.强阻尼波动方程的非协调混合有限元分析[J].河南师范大学学报(自然科学版),2016,44(2):22-28.
[12]刘倩,石东洋.双相滞热传导方程的一个非协调混合有限元方法[J].河南师范大学学报(自然科学版),2016,44(2):15-21.
[13]Pani A K.An H1-Galerkin mixed finite element methods for parabolic partial differential equations[J].SIAM J Numer Anal,1998,35(2):712-727.
[14]石东洋,唐启立,董晓靖.强阻尼波动方程的H1-Galerkin混合有限元超收敛分析[J].计算数学,2012,34(3):317-328.
[15]刁群,石东洋,张芳.Sobolev方程一个新的H1-Galerkin混合有限元分析[J].高校应用数学学报,2016,31(2):215-224.
[16]刁群,石东洋.拟线性粘弹性方程一个新的H1-Galerkin混合有限元分析[J].山东大学学报(理学版),2016,51(4):90-98.
[17]刁群,郭丽娟,王俊俊.非线性Sobolev方程低阶混合元方法的超收敛分析及外推[J].应用数学,2015,28(3):586-595.
[18]石东洋,董晓靖.非线性对流扩散方程的非协调EQrot1元解的渐近展开[J].计算数学,2012,40(3):1-5.
[19]Shi D Y,Wang J J.Unconditional superconvergence analysis of a linearized Galerkin FEM for nonlinear hyperbolic equations[J].Comput.Math Appl,2017,74:634-651.
[20]Hale J K.Ordinary Differential Equations[M].New York:Willey Inter Science,1969.
[21]王乐乐.若干偏微分方程的混合有限元方法研究[D].郑州:郑州大学,2017.
[22]Shi D Y,Wang P L,Zhao Y M.Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schr9dinger equation[J].Appl Math Letters,2014,38:129-134.
[23]石东洋,王芬玲,赵艳敏.非线性sine-Gordon方程的各向异性线性元高精度分析新模式[J].计算数学,2014,36(3):245-256.
[2]刘洋,李宏.四阶强阻尼波动方程的新的混合有限元方法[J].计算数学,2010,32(2):157-170.
[3]方志朝,刘洋,李宏.四阶强阻尼波动方程的混合控制体积法[J].计算数学,2011,33(4):409-422.
[4]张亚东,李新祥,石东洋.强阻尼波动方程的非协调有限元超收敛分析[J].山东大学学报(理学版),2014,49(5):28-35.
[5]罗振东.混合有限元方法基础及其应用[M].北京:科学出版社,2006.
[6]陈绍春,陈红如.二阶椭圆问题新的混合元格式[J].计算数学,2010,32(2):213-218.
[7]石东洋,李明浩.二阶问题一种新格式的高精度分析[J].应用数学学报,2014,37(1):45-58.
[8]石东洋,张亚东.抛物型方程一个新的非协调混合元超收敛性分析及外推[J].计算数学,2013,35(4):337-352.
[9]Shi D Y,Zhang Y D.High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equation[J].Appl Math Comput,2011,218(7):3176-3186.
[10]史艳华,石东洋.Sobolev方程新混合元方法的高精度分析[J].系统科学与数学,2014,34(4):452-463.
[11]毛凤梅,刁群.强阻尼波动方程的非协调混合有限元分析[J].河南师范大学学报(自然科学版),2016,44(2):22-28.
[12]刘倩,石东洋.双相滞热传导方程的一个非协调混合有限元方法[J].河南师范大学学报(自然科学版),2016,44(2):15-21.
[13]Pani A K.An H1-Galerkin mixed finite element methods for parabolic partial differential equations[J].SIAM J Numer Anal,1998,35(2):712-727.
[14]石东洋,唐启立,董晓靖.强阻尼波动方程的H1-Galerkin混合有限元超收敛分析[J].计算数学,2012,34(3):317-328.
[15]刁群,石东洋,张芳.Sobolev方程一个新的H1-Galerkin混合有限元分析[J].高校应用数学学报,2016,31(2):215-224.
[16]刁群,石东洋.拟线性粘弹性方程一个新的H1-Galerkin混合有限元分析[J].山东大学学报(理学版),2016,51(4):90-98.
[17]刁群,郭丽娟,王俊俊.非线性Sobolev方程低阶混合元方法的超收敛分析及外推[J].应用数学,2015,28(3):586-595.
[18]石东洋,董晓靖.非线性对流扩散方程的非协调EQrot1元解的渐近展开[J].计算数学,2012,40(3):1-5.
[19]Shi D Y,Wang J J.Unconditional superconvergence analysis of a linearized Galerkin FEM for nonlinear hyperbolic equations[J].Comput.Math Appl,2017,74:634-651.
[20]Hale J K.Ordinary Differential Equations[M].New York:Willey Inter Science,1969.
[21]王乐乐.若干偏微分方程的混合有限元方法研究[D].郑州:郑州大学,2017.
[22]Shi D Y,Wang P L,Zhao Y M.Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schr9dinger equation[J].Appl Math Letters,2014,38:129-134.
[23]石东洋,王芬玲,赵艳敏.非线性sine-Gordon方程的各向异性线性元高精度分析新模式[J].计算数学,2014,36(3):245-256.