Sobolev方程一个新的H~1-Galerkin混合有限元分析
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摘要
研究了Sobolev方程的H~1-Galerkin混合有限元方法.利用不完全双二次元Q_2~-和一阶BDFM元,建立了一个新的混合元模式,通过Bramble-Hilbert引理,证明了单元对应的插值算子具有的高精度结果.进一步,对于半离散和向后欧拉全离散格式,分别导出了原始变量u在H~1-模和中间变量p在H(div)-模意义下的超逼近性质.
In this paper,H~1-Galerkin mixed finite element method for Sobolev equation is studied.A new mixed finite element pattern is constructed using incomplete biquadratic element Q_2~- and first order BDFM element.Through Bramble-Hilbert lemma,high precision results of interpolation operators corresponding to unit are proved.Further,the superclose properties for the primitive variables u in H~1-norm and the intermediate variable p in H(div)-norm are obtained respectively in semi-discrete and the backward Euler fully discrete schemes.
In this paper,H~1-Galerkin mixed finite element method for Sobolev equation is studied.A new mixed finite element pattern is constructed using incomplete biquadratic element Q_2~- and first order BDFM element.Through Bramble-Hilbert lemma,high precision results of interpolation operators corresponding to unit are proved.Further,the superclose properties for the primitive variables u in H~1-norm and the intermediate variable p in H(div)-norm are obtained respectively in semi-discrete and the backward Euler fully discrete schemes.
引文
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[15]石东洋,唐启立,董晓靖.强阻尼波动方程的H~1-Galerkin混合有限元超收敛分析[J].计算数学,2012,34(3):317-328.
[16]石东洋,史艳华,王芬玲.四阶抛物方程H~1-Galerkin混合有限元方法的超逼近及最优误差估计[J].计算数学,2014,36(4):363-380.
[17]郭玲,陈焕贞.Sobolev方程的H~1-Galerkin混合有限元方法[J].系统科学与数学,2006,26(3):301-314.
[18]Shi Dongyang,Wang Haihong.Nonconforming H~1-Galerkin mixed FEM for Sobolev equtions on anisotropic meshes[J].Acta Math Appl Sin(English Ser),2009,25(2):335-344.
[19]李宏,季兆义,刘洋等.Sobolev方程的一类新型H~1-Galerkin混合元法[J].高等学校计算数学学报,2013,35(3):206-221.
[20]王金凤,刘洋,李宏等.Sobolev方程的基于H~1-Galerkin混合方法的新分裂格式[J].高等学校计算数学学报,2014,36(1):32-48.
[21]郝晓斌.非协调有限元的构造及其应用[D].郑州:郑州大学博士学位论文,2008.
[22]林群,严宁宁.高效有限元构造与分析[M].保定:河北大学出版社,1996.
[2]曹艳华.二维线性Sobolev方程广义差分法[J].计算数学,2005,27(3):243-256.
[3]石东洋,郝晓斌.Sobolev型方程各向异性Carey解的高精度分析[J].工程数学学报,2009,26(6):1021-1026.
[4]石东洋,任金城,郝晓斌.Sobolev型方程各向异性网格下Wilson元的高精度分析『J].高等学校计算数学学报,2009,31(2):169-180.
[5]何斯日古楞,李宏.Sobolev方程的时间间断Galerkin有限元方法[J].高校应用数学学报,2011,26(4):467-473.
[6]Shi Dongyang,Zhang Yadong.High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations[J].Appl Math Comput,2011,218(7):3176-3186.
[7]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.
[8]Shi Dongyang,Wang Haihong,Du Yuepeng.An anisotropic nonconforming finite element method for approximating a class of nonlinear Sobolev equations[J].J Comput Math,2009,27(2-3):299-314.
[9]Shi Dongyang,Wang Fenling,Zhao Yanmin.Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations[J].Acta Math Appl Sin(English Ser),2013,29(2):403-414.
[10]石东洋,王海红.非线性Sobolev方程的经济型差分-流线扩散非协调有限元法[J].数学物理学报,2013,33A(6):1148-1161.
[11]Pani A K.An H~1-Galerkin mixed finite element methods for parabolic partial differential equations[J].SIAM J Numer Anal,1998,35(2):721-727.
[12]王瑞文.双曲型积分微分方程H~1-Galerkin混合元法的误差估计[J].计算数学,2006,28(1):19-30.
[13]刘洋,李宏,何斯日古楞.伪双曲型积分-微分方程H~1-Galerkin混合元法误差估计[J].高等学校计算数学学报,2010,32(1):1-20.
[14]陈红斌,刘晓奇,徐大.粘弹性双曲型方程的H~1-Galerkin混合有限元方法[J].高等学校计算数学学报,2011,33(3):279-288.
[15]石东洋,唐启立,董晓靖.强阻尼波动方程的H~1-Galerkin混合有限元超收敛分析[J].计算数学,2012,34(3):317-328.
[16]石东洋,史艳华,王芬玲.四阶抛物方程H~1-Galerkin混合有限元方法的超逼近及最优误差估计[J].计算数学,2014,36(4):363-380.
[17]郭玲,陈焕贞.Sobolev方程的H~1-Galerkin混合有限元方法[J].系统科学与数学,2006,26(3):301-314.
[18]Shi Dongyang,Wang Haihong.Nonconforming H~1-Galerkin mixed FEM for Sobolev equtions on anisotropic meshes[J].Acta Math Appl Sin(English Ser),2009,25(2):335-344.
[19]李宏,季兆义,刘洋等.Sobolev方程的一类新型H~1-Galerkin混合元法[J].高等学校计算数学学报,2013,35(3):206-221.
[20]王金凤,刘洋,李宏等.Sobolev方程的基于H~1-Galerkin混合方法的新分裂格式[J].高等学校计算数学学报,2014,36(1):32-48.
[21]郝晓斌.非协调有限元的构造及其应用[D].郑州:郑州大学博士学位论文,2008.
[22]林群,严宁宁.高效有限元构造与分析[M].保定:河北大学出版社,1996.