两类非线性方程的混合元方法研究
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摘要
本论文由四部分组成.
第一部分主要利用双线性元和零阶R—T元,对非线性sine-Gordon方程构造了一个新的协调矩形混合元变分形式,基于积分恒等式技巧,导数转移及插值算子的特性,给出了在半离散格式下相关变量的超逼近性质.同时使用插值后处理技术得到了相应的整体超收敛结果.
第二部分主要对非线性sine-Gordon方程提出一个新的非协调混合有限元逼近格式,基于EQ_1~(rot)非协调元有两个特殊性质：(a)当精确解属于H3(Ω)时,其相容误差为O(h2)阶比它的插值误差高一阶；(b)插值算子与Ritz投影算子等价,再结合零阶R—T元的高精度结果,导出了半离散格式精确解u在H1模意义下和中间变量p在L2模意义下二阶超逼近及超收敛.最后,讨论了一个二阶全离散逼近格式,得到了精确解u在H1模意义下和流量p在L2模意义下的最优误差估计.
第三部分对非线性sine-Gordon方程利用协调线性三角形元构造了一个新的H1—Galerkin混合有限元格式,利用上述的方法和技巧,给出了在半离散及全离散格式下解的超逼近性质及超收敛结果.
第四部分主要利用双线性元和零阶R—T元及EQ_1~(rot)元对非线性伪双曲方程初边值问题分别构造了新的协调及非协调混合元格式,在抛弃传统有限元分析中不可缺少的工具—Ritz投影的前提下,借助平均值技巧,积分恒等式和插值后处理技术,给出了解的超逼近性质及整体超收敛结果.
This dissertation includes four parts.
In the first part, a new conforming rectangular mixed finite element viriational form is constructed for sine-Gordon equations with the bilinear element and zero order R-T element. By using integral identity techniques, derivative transfer and interpolation operator's characteristics, we get the superclose properties of the in semi-discrete scheme. At the same time, by virtue of the technique of interpolation post-processing technique, the global superconvergence results are obtained.
In the second part, a new nonconforming mixed finite element approximation scheme for nonlinear sine-Gordon equations is proposed. By use of two special properties of this element:(a) the consistency error is of order O(h2) one order higher than its interpolation error O(h), when the exact solution belongs to H3(Ω);(b) the interpolation operator is equivalent to its Ritz-pr ejection operator,the superclose properties and superconvergences of exact solution u in H1-norm and intermediate variable p in L2-norm are deduced for semi-discrete scheme, and the optimal order error estimates are obtained for a two order fully-discrete scheme.
In the third part, by use of the given estimates of conforming linear trianglar element, a new H1-Galerkin mixed finite element viriational form is constructed for sine-Gordon equations. With the help of the aforementioned methods and skills, we get the superclose property and superconvergence in both semi-discrete form and fully discrete form.
In the last part, a new conforming and nonconforming mixed finite element methods are proposed for Nonlinear pseudo-hyperbolic equations respectively, without using Ritz projection, which is an indispensable tool in the traditional finite element analysise, the superclose and the global superconvergence are derived directly through interpolation operater, mean-value tricle and post-processing technique.
第一部分主要利用双线性元和零阶R—T元,对非线性sine-Gordon方程构造了一个新的协调矩形混合元变分形式,基于积分恒等式技巧,导数转移及插值算子的特性,给出了在半离散格式下相关变量的超逼近性质.同时使用插值后处理技术得到了相应的整体超收敛结果.
第二部分主要对非线性sine-Gordon方程提出一个新的非协调混合有限元逼近格式,基于EQ_1~(rot)非协调元有两个特殊性质：(a)当精确解属于H3(Ω)时,其相容误差为O(h2)阶比它的插值误差高一阶；(b)插值算子与Ritz投影算子等价,再结合零阶R—T元的高精度结果,导出了半离散格式精确解u在H1模意义下和中间变量p在L2模意义下二阶超逼近及超收敛.最后,讨论了一个二阶全离散逼近格式,得到了精确解u在H1模意义下和流量p在L2模意义下的最优误差估计.
第三部分对非线性sine-Gordon方程利用协调线性三角形元构造了一个新的H1—Galerkin混合有限元格式,利用上述的方法和技巧,给出了在半离散及全离散格式下解的超逼近性质及超收敛结果.
第四部分主要利用双线性元和零阶R—T元及EQ_1~(rot)元对非线性伪双曲方程初边值问题分别构造了新的协调及非协调混合元格式,在抛弃传统有限元分析中不可缺少的工具—Ritz投影的前提下,借助平均值技巧,积分恒等式和插值后处理技术,给出了解的超逼近性质及整体超收敛结果.
This dissertation includes four parts.
In the first part, a new conforming rectangular mixed finite element viriational form is constructed for sine-Gordon equations with the bilinear element and zero order R-T element. By using integral identity techniques, derivative transfer and interpolation operator's characteristics, we get the superclose properties of the in semi-discrete scheme. At the same time, by virtue of the technique of interpolation post-processing technique, the global superconvergence results are obtained.
In the second part, a new nonconforming mixed finite element approximation scheme for nonlinear sine-Gordon equations is proposed. By use of two special properties of this element:(a) the consistency error is of order O(h2) one order higher than its interpolation error O(h), when the exact solution belongs to H3(Ω);(b) the interpolation operator is equivalent to its Ritz-pr ejection operator,the superclose properties and superconvergences of exact solution u in H1-norm and intermediate variable p in L2-norm are deduced for semi-discrete scheme, and the optimal order error estimates are obtained for a two order fully-discrete scheme.
In the third part, by use of the given estimates of conforming linear trianglar element, a new H1-Galerkin mixed finite element viriational form is constructed for sine-Gordon equations. With the help of the aforementioned methods and skills, we get the superclose property and superconvergence in both semi-discrete form and fully discrete form.
In the last part, a new conforming and nonconforming mixed finite element methods are proposed for Nonlinear pseudo-hyperbolic equations respectively, without using Ritz projection, which is an indispensable tool in the traditional finite element analysise, the superclose and the global superconvergence are derived directly through interpolation operater, mean-value tricle and post-processing technique.
引文
[1]I.Babuska. Error-bounds for finite element method[J]. Numer. Math.,1971,16(5):322-333.
[2]F.Brezzi, M.Fortin. Mixed and Hybrid Finite Element Methods[M]. Springer-Verlag.,1991.
[3]R.S.Falk, J.E.Osborn. Error estimates for mixed methods[J]. RAIRO Anal. Numer.,1980, 14(3):249-277.
[4]徐秋滨,张鲁明.二维广义非线性sine-Gordon方程的一个ADI格式[J].应用数学学报,2007,30(5)：836-846.
[5]Liang Zong-qi. The global solution and numerical computation of the generalized nonlinear Sine-Gordon Equation[J]. Appl. Math.,2003,16(4):40-49.
[6]Zhou Sheng-fan. Dimension of the Global Attractor for the Damped sine-Gordon Equa-tion[J]. Acta Math. Sin.,1996,39(5):597-601.
[7]盛平兴.广义sine-Gordon方程的混沌与湍流[J].应用数学学报,2005,28(3)：836-846.
[8]黄德斌,刘曾荣,郑永爱.sine-Gordon方程中混沌跳跃行为的机制[J].力学学报,2002,34(2)：238-247.
[9]Zhang Fei, Luis Vazquez. Two energy conserving numerical schemes for the sine-Gordon equation [J]. Appl. Math. Comput.,1991,45:17-30.
[10]张鲁明sine-Gordon方程的能量守恒差分格式[J].石油大学学报,1999,23(2)：100-103.
[11]张鲁明,常谦顺sine-Gordon方程的一个守恒的九点格式[J].应用数学,1999,12(3)：30-35.
[12]徐秋滨.有阻尼的sine-Gordon方程的几个差分格式[J].南京航空航天大学学报,2005,2(1)：35-41.
[13]徐秋滨,常谦顺.广义线性sine-Gordon方程的两个隐式格式[J].应用数学学报,2007,20(3)：263-271.
[14]Lu Fa-lin, Chen Chang-yuan, Sun Dong-sheng. Bound states of Klein-Gordon equation for double ring-shaped oscillator scalar and vector potentials [J].Chinese Phys.B.2005,14(3): 463-467.
[15]Zhang Xue-ao, Chen Ke, Duan Zheng-lu. Bound states of Klein-Gordon equation and Dirac equation for ringshaped non-spherical oscillator scalar and vector potentials[J]. Chinese Phys.B.2005,12(4):322-326.
[16]Zheng Qiang, Yue Ping, Gong Lun-xun. New exact solutions of nonlinear Klein-Gordon equation[J]. Chinese Phys.B.2006,14(1):135-141.
[17]He Hong-sheng, Chen Jiang, Yang Kong-qing. Exact solutions for the coupled Klein-Gordon Schrodinger equations using the extended F-expansion method[J]. Chinese Phys.B. 2005,12(4):100-107.
[18]樊明智,王芬玲,石东洋.非线性sine-Gordon方程的双线性元分析及外推[J].数学的实践与认识,2012,42(11)：205-213.
[19]梁洪亮,赵树理.一类非线性广义sine-Gordon方程的有限元超收敛分析[J].四川师范大学学报.2012,35(5)：610-614.
[20]王芬玲,石东洋.非线性sine-Gordon方程Hermite型有限元新的超收敛分析[J].应用数学学报.2012,35(5)：777-787.
[21]石东洋,张斐然.Sine-Gordon方程的一类低阶非协调有限元分析[J].计算数学,2011,33(3)：289-297.
[22]Liu Yang, Li Hong. Numerical solutions of H1-Galerkin mixed finite element method for a damped Sine-Gordon Equation[J]. Appl. Math.,2009,22(3):579-588.
[23]C. V. Pao. A mixed initial boundary-value problem arising in neurophysiology[J]. J. Math. Anal. Appl.,1975,52(1):105-119.
[24]K. C. Thmas, J. Elloson. Existence uniqueness and stability of solution of a class of nonlinear partial differential equations [J]. J. Math. Anal. Appl.,1975,51(1):1-32.
[25]G. Ponce. Global existence of small of solutions to a class of nonlinear evolution equations[J]. Nonlinear Anal.,1985,9(5),399-418.
[26]A K Pani, R K Sinha, A K Otta. An H1-Galerkin mixed method for second order hyperbolic equation[J]. Int. J. Numer. Anal. Model.,2004,1(2):111-129.
[27]Chen Guo-wang, Yang Zhi-jian. Inital value problem for a class of nonlinear Pseudo-hyperbolic equations[J]. ACTA,Math. Appl. Sin.,1993,9(2):166-173.
[28]L.Davis Paul. A quasilinear hypebolic and ralated third-order equations[J]. J. Math. Anal. Appl.,1975,51(9):596-606.
[29]Liu Yang, Li Hong. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equa-tions[J]. Appl. Math. Comput,2009,212:446-457.
[30]Liu Yang, Li Hong. A new mixed finite element methods for pseudo-hyperbolic equations [J]. Appl. Math. Comput.,2010,23(1):150-157.
[31]Guo Hui, Rui Hong-xing. A remark on least-squares Galerkin procedures for pseudo hyperbolic-equations[J]. J. Comput. Appl. Math.,2009,229(1):108-119.
[32]Guo Hui, Rui Hong-xing. Least-squares Galerkin procedures for pseudo hyperbolic-equations[J]. J. Comput. Appl. Math.,2007,189(1):425-429.
[33]Shi Dong-yang, Zhang Ya-Dong. An H1-Galerkin noncoforming mixed finite element method for pseudo Hyperbolic-equations [J]. Math. Appl.,2011,24(3):448-455.
[34]Liu Yang, Li Hong. A new mixed finite element method for fourth-order heavy damping wave equation[J]. Math. Numer. Sin.,2010,32(2):157-170.
[35]张志跃.一类广义神经传播方程的有限元方法及其数值分析[J].数学物理学报,2001,21A：647-654.
[36]P.G. Ciarlet. The Finite Element Method For Elliptic Problems[M]. North-Holland:Ams-terdam,1978.
[37]王烈衡,许学军.有限元方法的数学基础[M].北京：科学出版社,2004.
[38]石钟慈,王鸣.有限元方法[M].北京：科学出版社,2010.
[39]罗振东.混合有限元法基础及其应用[M].北京：科学出版社,2006.
[40]R.A. Adams. Sobolev Spaces[M]. New York:Academic Press,1975.
[41]石东洋,谢萍丽.Sobolev方程的一类各向异性非协调有限元逼近[J].系统科学与数学,2006,29(1)：116-128.
[42]林群,严宁宁.高校有限元构造与分析[M].河北：河北大学出版社,1996.
[43]Lin Qun, Tobiska L, Zhou Ai-hui. Superconvergence and extrapolation of nonconformimg low order finite elements applied to the Poisson equation[J]. IMA Journal Numerical Analysis, 2005,25(1):160-181.
[44]Shi Dong-yang, Mao Shi-peng, Chen Shao-chun. An anisotropic nonconforming finite element with some superconvergence results[J]. J. Comput. Math.,2005,23(3):261-274.
[45]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[J].工程数学学报,2011,28(2)：231-236.
[46]Shi Dong-yang, Zhang Ya-dong. High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equation[J]. Appl. Math. Compu.,2011,218(7):3176-3186.
[47]石东洋,张熠然.非定常Stokes问题的矩形Crouzeix-Raviart型各向异性非协调元变网格方法[J].数学物理学报,2006,26A(5)：659-670.
[48]刘洋,李宏,和斯日古楞.伪双曲型积分-微分方程的H1-Galerkin混合元误差估计[J].高等学校计算数学学报,2010,32(1)：1-20.
[49]石东洋,梁慧.各向异性网格下线性三角形元的超收敛分析[J].工程数学物理学报,2007,24(3)：487-493.
[50]Lin Qun, Lin Jia-fu. Finite Element Methods Accuracy And Improvement[M]. Beijing:Science Press,2006.
[51]Yan Ning-ning. Superconvergence Analysis And A Posteriori Error Estimation In finite Element Methods[M]. Beijing:Science Press,2008.
[52]Shi Dong-yang, Gong Wei. High accuracy analysis of fully discrete Galerkin approximation for parabolic equetions on anistropic meshes[J]. Acta Math, sci.,2009,29a(4):898-911.
[2]F.Brezzi, M.Fortin. Mixed and Hybrid Finite Element Methods[M]. Springer-Verlag.,1991.
[3]R.S.Falk, J.E.Osborn. Error estimates for mixed methods[J]. RAIRO Anal. Numer.,1980, 14(3):249-277.
[4]徐秋滨,张鲁明.二维广义非线性sine-Gordon方程的一个ADI格式[J].应用数学学报,2007,30(5)：836-846.
[5]Liang Zong-qi. The global solution and numerical computation of the generalized nonlinear Sine-Gordon Equation[J]. Appl. Math.,2003,16(4):40-49.
[6]Zhou Sheng-fan. Dimension of the Global Attractor for the Damped sine-Gordon Equa-tion[J]. Acta Math. Sin.,1996,39(5):597-601.
[7]盛平兴.广义sine-Gordon方程的混沌与湍流[J].应用数学学报,2005,28(3)：836-846.
[8]黄德斌,刘曾荣,郑永爱.sine-Gordon方程中混沌跳跃行为的机制[J].力学学报,2002,34(2)：238-247.
[9]Zhang Fei, Luis Vazquez. Two energy conserving numerical schemes for the sine-Gordon equation [J]. Appl. Math. Comput.,1991,45:17-30.
[10]张鲁明sine-Gordon方程的能量守恒差分格式[J].石油大学学报,1999,23(2)：100-103.
[11]张鲁明,常谦顺sine-Gordon方程的一个守恒的九点格式[J].应用数学,1999,12(3)：30-35.
[12]徐秋滨.有阻尼的sine-Gordon方程的几个差分格式[J].南京航空航天大学学报,2005,2(1)：35-41.
[13]徐秋滨,常谦顺.广义线性sine-Gordon方程的两个隐式格式[J].应用数学学报,2007,20(3)：263-271.
[14]Lu Fa-lin, Chen Chang-yuan, Sun Dong-sheng. Bound states of Klein-Gordon equation for double ring-shaped oscillator scalar and vector potentials [J].Chinese Phys.B.2005,14(3): 463-467.
[15]Zhang Xue-ao, Chen Ke, Duan Zheng-lu. Bound states of Klein-Gordon equation and Dirac equation for ringshaped non-spherical oscillator scalar and vector potentials[J]. Chinese Phys.B.2005,12(4):322-326.
[16]Zheng Qiang, Yue Ping, Gong Lun-xun. New exact solutions of nonlinear Klein-Gordon equation[J]. Chinese Phys.B.2006,14(1):135-141.
[17]He Hong-sheng, Chen Jiang, Yang Kong-qing. Exact solutions for the coupled Klein-Gordon Schrodinger equations using the extended F-expansion method[J]. Chinese Phys.B. 2005,12(4):100-107.
[18]樊明智,王芬玲,石东洋.非线性sine-Gordon方程的双线性元分析及外推[J].数学的实践与认识,2012,42(11)：205-213.
[19]梁洪亮,赵树理.一类非线性广义sine-Gordon方程的有限元超收敛分析[J].四川师范大学学报.2012,35(5)：610-614.
[20]王芬玲,石东洋.非线性sine-Gordon方程Hermite型有限元新的超收敛分析[J].应用数学学报.2012,35(5)：777-787.
[21]石东洋,张斐然.Sine-Gordon方程的一类低阶非协调有限元分析[J].计算数学,2011,33(3)：289-297.
[22]Liu Yang, Li Hong. Numerical solutions of H1-Galerkin mixed finite element method for a damped Sine-Gordon Equation[J]. Appl. Math.,2009,22(3):579-588.
[23]C. V. Pao. A mixed initial boundary-value problem arising in neurophysiology[J]. J. Math. Anal. Appl.,1975,52(1):105-119.
[24]K. C. Thmas, J. Elloson. Existence uniqueness and stability of solution of a class of nonlinear partial differential equations [J]. J. Math. Anal. Appl.,1975,51(1):1-32.
[25]G. Ponce. Global existence of small of solutions to a class of nonlinear evolution equations[J]. Nonlinear Anal.,1985,9(5),399-418.
[26]A K Pani, R K Sinha, A K Otta. An H1-Galerkin mixed method for second order hyperbolic equation[J]. Int. J. Numer. Anal. Model.,2004,1(2):111-129.
[27]Chen Guo-wang, Yang Zhi-jian. Inital value problem for a class of nonlinear Pseudo-hyperbolic equations[J]. ACTA,Math. Appl. Sin.,1993,9(2):166-173.
[28]L.Davis Paul. A quasilinear hypebolic and ralated third-order equations[J]. J. Math. Anal. Appl.,1975,51(9):596-606.
[29]Liu Yang, Li Hong. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equa-tions[J]. Appl. Math. Comput,2009,212:446-457.
[30]Liu Yang, Li Hong. A new mixed finite element methods for pseudo-hyperbolic equations [J]. Appl. Math. Comput.,2010,23(1):150-157.
[31]Guo Hui, Rui Hong-xing. A remark on least-squares Galerkin procedures for pseudo hyperbolic-equations[J]. J. Comput. Appl. Math.,2009,229(1):108-119.
[32]Guo Hui, Rui Hong-xing. Least-squares Galerkin procedures for pseudo hyperbolic-equations[J]. J. Comput. Appl. Math.,2007,189(1):425-429.
[33]Shi Dong-yang, Zhang Ya-Dong. An H1-Galerkin noncoforming mixed finite element method for pseudo Hyperbolic-equations [J]. Math. Appl.,2011,24(3):448-455.
[34]Liu Yang, Li Hong. A new mixed finite element method for fourth-order heavy damping wave equation[J]. Math. Numer. Sin.,2010,32(2):157-170.
[35]张志跃.一类广义神经传播方程的有限元方法及其数值分析[J].数学物理学报,2001,21A：647-654.
[36]P.G. Ciarlet. The Finite Element Method For Elliptic Problems[M]. North-Holland:Ams-terdam,1978.
[37]王烈衡,许学军.有限元方法的数学基础[M].北京：科学出版社,2004.
[38]石钟慈,王鸣.有限元方法[M].北京：科学出版社,2010.
[39]罗振东.混合有限元法基础及其应用[M].北京：科学出版社,2006.
[40]R.A. Adams. Sobolev Spaces[M]. New York:Academic Press,1975.
[41]石东洋,谢萍丽.Sobolev方程的一类各向异性非协调有限元逼近[J].系统科学与数学,2006,29(1)：116-128.
[42]林群,严宁宁.高校有限元构造与分析[M].河北：河北大学出版社,1996.
[43]Lin Qun, Tobiska L, Zhou Ai-hui. Superconvergence and extrapolation of nonconformimg low order finite elements applied to the Poisson equation[J]. IMA Journal Numerical Analysis, 2005,25(1):160-181.
[44]Shi Dong-yang, Mao Shi-peng, Chen Shao-chun. An anisotropic nonconforming finite element with some superconvergence results[J]. J. Comput. Math.,2005,23(3):261-274.
[45]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[J].工程数学学报,2011,28(2)：231-236.
[46]Shi Dong-yang, Zhang Ya-dong. High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equation[J]. Appl. Math. Compu.,2011,218(7):3176-3186.
[47]石东洋,张熠然.非定常Stokes问题的矩形Crouzeix-Raviart型各向异性非协调元变网格方法[J].数学物理学报,2006,26A(5)：659-670.
[48]刘洋,李宏,和斯日古楞.伪双曲型积分-微分方程的H1-Galerkin混合元误差估计[J].高等学校计算数学学报,2010,32(1)：1-20.
[49]石东洋,梁慧.各向异性网格下线性三角形元的超收敛分析[J].工程数学物理学报,2007,24(3)：487-493.
[50]Lin Qun, Lin Jia-fu. Finite Element Methods Accuracy And Improvement[M]. Beijing:Science Press,2006.
[51]Yan Ning-ning. Superconvergence Analysis And A Posteriori Error Estimation In finite Element Methods[M]. Beijing:Science Press,2008.
[52]Shi Dong-yang, Gong Wei. High accuracy analysis of fully discrete Galerkin approximation for parabolic equetions on anistropic meshes[J]. Acta Math, sci.,2009,29a(4):898-911.
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