对流扩散方程及强阻尼波动方程的高精度分析
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摘要
本论文包括两部分.第一部分主要研究发展型非线性对流扩散方程的双线性元及非协调EQ1rot元逼近,给出了L2(Ω)模意义下的最优ε一致收敛性结果.同时根据Bramble-Hilbert引理分别导出了高精度的积分恒等式,并由此导出了一些新的渐近展开式.利用外推技巧得到了具有三阶精度的近似解.第二部分主要研究了强阻尼波动方程的H1-Galerkin混合有限元方法的超收敛性.借助于协调线性三角形元已有的分析估计式,直接利用插值算子代替原始变量μ的Ritz投影和应力变量p的Ritz-Volterra投影.对半离散和全离散格式,得到了μ在H1(Ω)模和p在H(div;Ω)模意义下比以往文献高一阶的超逼近和超收敛结果.
This dissertation mainly includes two parts. In the first part, the optimal ε uniform convergent results of bilinear conforming finite element and nonconforming finite element EQ1rot are obtained under L2(Ω) norm for the time-dependent nonlinear advection-diffusion equations. Based on Bramble-Hilbert lemma, higher accuracy integral identities and some new asymptotic expansions are derived. Moreover, we derive the related approximate solutions with third order by use of the extrapolation technique. In the second part, the superconvergence analysis of H1-Galerkin mixed finite element method for strongly damped wave equations is studied. By virtue of the interpolation technique operator instead of Ritz projection of the original variable u and Ritz-Volterra projection of the stress variable p, the superclose and superconvergence results in in(Q) norm for u and H (div;Ω) norm for p for both semidiscrete and fully discrete schemes are derived through applying some error estimates of conforming linear triangular finite element.
This dissertation mainly includes two parts. In the first part, the optimal ε uniform convergent results of bilinear conforming finite element and nonconforming finite element EQ1rot are obtained under L2(Ω) norm for the time-dependent nonlinear advection-diffusion equations. Based on Bramble-Hilbert lemma, higher accuracy integral identities and some new asymptotic expansions are derived. Moreover, we derive the related approximate solutions with third order by use of the extrapolation technique. In the second part, the superconvergence analysis of H1-Galerkin mixed finite element method for strongly damped wave equations is studied. By virtue of the interpolation technique operator instead of Ritz projection of the original variable u and Ritz-Volterra projection of the stress variable p, the superclose and superconvergence results in in(Q) norm for u and H (div;Ω) norm for p for both semidiscrete and fully discrete schemes are derived through applying some error estimates of conforming linear triangular finite element.
引文
[1]Johnson C Streamline diffusion methods for problems in fluid mechanics, in Finite Elements in fluids[M]. VI, Wiley, New York,1986.
[2]Yang Danping. Analysis of least-squares mixed finite element methods for nonlinear non-stationary convection-diffusion problem[J]. Math. Comp.,2000,69:929-936.
[3]Dawson C.N., Russell T.F., Wheeler M.F.. Some improved error estimates for the modified method of characteristics [J]. SI AM J. Numer. Anal.,1989,26:1487-1512.
[4]Douglas Jr.J., Russell T.F.. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures [J]. SIAM J. Numer. Anal.,1982,19:871-885.
[5]Guo Ling, Chen Huanzhen. An expanded characteristic-mixed finite element method for a convection-dominated transport problem[J]. J. Comput. Math.,2005,23:479-490.
[6]朱国庆,陈绍春.一个新非协调单元对扩散对流反应方程的应用[J].数学的实践与认识,2010,40(8)：126-131.
[7]Wang Kaixin, Wang Hong. An optimal-order error estimate to the modified method of char-acteristics for a degenerate convection-diffusion equation[J]. Int. J. Numer. Anal. Model., 2009,6:271-231.
[8]Lin Qun, Wang Hong, Zhang Shuhua. Uniform optimal-order estimates for finite element methods for advection-diffusion equations [J]. J. Sys. Sci. Complex.,2009,22(4):555-559.
[9]Fairweather G., Lin Qun, Lin Yanping, Wang Junping, Zhang Shuhua. Asymptotic expan-sions and Richardson extrapolation of approximate solutions for second order elliptic prob-lems on rectangular domains by mixed finite element methods[J]. SIAM J. Numer. Anal., 2006,44:1122-1149.
[10]石东洋，任金城Stokes f司题Q1-P1混合元外推方法[J].数学的实践与认识,2008,38(17)：66-75.
[11]黄云清,林群.抛物型方程有限元解的渐近展式及超收敛[J].系统科学与数学,1991,11(4)：32-335.
[12]王军平,林群.有限元方法的渐近展开及外推[J].系统科学与数学,1985,5(2)：114-120.
[13]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.
[14]石东洋,郝晓斌Sobolev型方程各向异性Carey元解的高精度分析[J].工程数学学报,2009，26(6)：1021-1026.
[15]Pani A.K.. An H1-Galerkin mixed finite element method for parabolic partial differential equations[J]. SIAM J. Numer. Anal.,1998,35:712-727.
[16]Guo Ling, Chen Huanzhen. H1-Galerkin mixed finite element method for the regularized long wave equation [J]. Computing,2006,77:205-221.
[17]Liu Yang, Li Hong. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equa-tions[J]. Appl. Math. Comp.,2009,212:446-457.
[18]Shi Dongyang, Wang Haihong. Noncorforming H1-Galerkin mixed FEM for Sobolev equa-tions on anisotropic meshes[J]. Acta Math. Appl. Sinica,2009,25:335-344.
[19]Pani A.K., Fairweather G.. H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J]. IMA J. Numer. Anal,2002,22:231-252.
[20]陈红斌,徐大,刘晓奇.非线性抛物型偏积分微分方程的H1-Galerkin混合有限元方法[J]应用数学学报，2008，31(4)：702-712.
[21]Shi Dongyang, Wang Haihong. An H1-Galerkin nonconforming mixed finite element method for integro-differential equations of parabolic type[J]. J. Math. Res. Exposition,2009,29(5): 871-881.
[22]Zhou Zhaojie. An H1-Galerkin mixed finite element methods for a class of heat transport equations[J]. Appl. Math. Model..2010,34:2414-2425.
[23]郭玲,陈焕贞Sobolev方程的H1-Galerkin混合有限元法[J].系统科学与数学,2006,26(3)：301-314.
[24]刘洋.李宏.四阶强阻尼波动方程的新混合元方法[J].计算数学,2010.32(2)：157-170.
[25]Ciarlet P.G.. The Finite Element Method for Elliptic Problems[M]. North-Holland:Amster-dam,1978.
[26]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.
[27]石钟慈,王鸣.有限元方法[M].北京：科学出版社,2010.
[28]罗振东,混合有限元法基础及其应用[M].北京：科学出版社,2006.
[29]Adams R.A. Sobolev Spaces[M]. New York:Academic Press,1975.
[30]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.
[31]Lin Qun, Lin Jiafu. Finite Element Methods:Accuracy and Improvement [M]. Beijing:Sci-ence Press,2006.
[32]Yan Ningning. Superconvergence Analysis and a Posteriori Error Estimation in Finite Ele-ment Methods[M]. Beijing:Science Press,2008.
[33]Wang Hong. An optimal-order error estimate for an ELLAM scheme for two-dimensional linear advection-diffusion equation[J]. SIAM J. Numer. Anal.,2000,37:1338-1368.
[34]Bause M. Knabner P.. Uniform error analysis for Lagrange-Galerkin approximation of con-vection domainated problems [J]. SIAM J. Numer. Anal.,2002,39:1954-1984.
[35]Shi Dongyang, Wang Haihong, Du Yuepeng. An anisotropic noncomforming finite element for approximating a class of nonlinear Sobolev equations [J]. J. Comput. Math.,2009,27(2-3):299-314.
[36]方志朝,李宏,刘洋.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.
[37]Park C, Sheen D.W.. P1 nonconforming quadrilateral finite element methods for second-order elliptic problems [J]. SIAM J. Numer. Anal.,2003,41:624-640
[38]Lin Qun, Tobiska L.. Zhou Aihui. Superconvergence and extrapolation of nonconforming low-order elements applied to the Poisson equation [J]. IMA J. Numer. Anal.,2005,25:160-181.
[39]Rannacher R., Turek S.. Simple nonconforming quadrilateral stokes element [J]. Numer. Meth. for PDE,1992,8:97-111.
[40]胡俊,满红英,石钟慈.带约束非协调旋转Q1元在Stokes和平面弹性问题中的应用[J].计算数学,2005,27(3)：311-324.
[41]Hu Jun, Shi Zhongci. Constrained quadrilateral nonconforming rotated Q1-element[J]. J. Comput. Math.,2005,23(5):561-586.
[42]Liu Huipo, Yan Ningning. Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations [J]. Adv. Comput. Math.,2008,29:375-392.
[43]Apel T., Nicaise S., Schoberl J.. Crouzeix-Raviart type finite element on anisotropic meshes[J]. Numer. Math.,2001,89:193-293.
[44]Shi Dongyang, Zhang Buying. High accuracy analysis for the finite element method for nonlinear viscoelastic wave equations with nonlinear boundary conditions [J]. J. Syst. Sci. Complex.,2011,24:795-802.
[45]Shi Dongyang, Zhang Buying. High accuracy analysis of the finite elemnet method for nonlinear parabolic integro-differential equations with nonlinear boundary conditions[J]数学进展,2009,38(6)：715-722.
[46]石东洋,王慧敏.非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近[J].数学物理学报,2010,30A(4)：1018-1029.
[47]Knobloch P., Tobiska L.. The P1mod element:a new nonconforming finite elemnet for convection-diffusion problems[J]. SIAM J. Numer. Anal.,2003,41:436-456.
[48]石东洋,梁慧.各向异性网格下线性三角形元的超收敛性分析[J].工程数学学报，2007,24(3)：487-493.
[2]Yang Danping. Analysis of least-squares mixed finite element methods for nonlinear non-stationary convection-diffusion problem[J]. Math. Comp.,2000,69:929-936.
[3]Dawson C.N., Russell T.F., Wheeler M.F.. Some improved error estimates for the modified method of characteristics [J]. SI AM J. Numer. Anal.,1989,26:1487-1512.
[4]Douglas Jr.J., Russell T.F.. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures [J]. SIAM J. Numer. Anal.,1982,19:871-885.
[5]Guo Ling, Chen Huanzhen. An expanded characteristic-mixed finite element method for a convection-dominated transport problem[J]. J. Comput. Math.,2005,23:479-490.
[6]朱国庆,陈绍春.一个新非协调单元对扩散对流反应方程的应用[J].数学的实践与认识,2010,40(8)：126-131.
[7]Wang Kaixin, Wang Hong. An optimal-order error estimate to the modified method of char-acteristics for a degenerate convection-diffusion equation[J]. Int. J. Numer. Anal. Model., 2009,6:271-231.
[8]Lin Qun, Wang Hong, Zhang Shuhua. Uniform optimal-order estimates for finite element methods for advection-diffusion equations [J]. J. Sys. Sci. Complex.,2009,22(4):555-559.
[9]Fairweather G., Lin Qun, Lin Yanping, Wang Junping, Zhang Shuhua. Asymptotic expan-sions and Richardson extrapolation of approximate solutions for second order elliptic prob-lems on rectangular domains by mixed finite element methods[J]. SIAM J. Numer. Anal., 2006,44:1122-1149.
[10]石东洋，任金城Stokes f司题Q1-P1混合元外推方法[J].数学的实践与认识,2008,38(17)：66-75.
[11]黄云清,林群.抛物型方程有限元解的渐近展式及超收敛[J].系统科学与数学,1991,11(4)：32-335.
[12]王军平,林群.有限元方法的渐近展开及外推[J].系统科学与数学,1985,5(2)：114-120.
[13]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.
[14]石东洋,郝晓斌Sobolev型方程各向异性Carey元解的高精度分析[J].工程数学学报,2009，26(6)：1021-1026.
[15]Pani A.K.. An H1-Galerkin mixed finite element method for parabolic partial differential equations[J]. SIAM J. Numer. Anal.,1998,35:712-727.
[16]Guo Ling, Chen Huanzhen. H1-Galerkin mixed finite element method for the regularized long wave equation [J]. Computing,2006,77:205-221.
[17]Liu Yang, Li Hong. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equa-tions[J]. Appl. Math. Comp.,2009,212:446-457.
[18]Shi Dongyang, Wang Haihong. Noncorforming H1-Galerkin mixed FEM for Sobolev equa-tions on anisotropic meshes[J]. Acta Math. Appl. Sinica,2009,25:335-344.
[19]Pani A.K., Fairweather G.. H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J]. IMA J. Numer. Anal,2002,22:231-252.
[20]陈红斌,徐大,刘晓奇.非线性抛物型偏积分微分方程的H1-Galerkin混合有限元方法[J]应用数学学报，2008，31(4)：702-712.
[21]Shi Dongyang, Wang Haihong. An H1-Galerkin nonconforming mixed finite element method for integro-differential equations of parabolic type[J]. J. Math. Res. Exposition,2009,29(5): 871-881.
[22]Zhou Zhaojie. An H1-Galerkin mixed finite element methods for a class of heat transport equations[J]. Appl. Math. Model..2010,34:2414-2425.
[23]郭玲,陈焕贞Sobolev方程的H1-Galerkin混合有限元法[J].系统科学与数学,2006,26(3)：301-314.
[24]刘洋.李宏.四阶强阻尼波动方程的新混合元方法[J].计算数学,2010.32(2)：157-170.
[25]Ciarlet P.G.. The Finite Element Method for Elliptic Problems[M]. North-Holland:Amster-dam,1978.
[26]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.
[27]石钟慈,王鸣.有限元方法[M].北京：科学出版社,2010.
[28]罗振东,混合有限元法基础及其应用[M].北京：科学出版社,2006.
[29]Adams R.A. Sobolev Spaces[M]. New York:Academic Press,1975.
[30]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.
[31]Lin Qun, Lin Jiafu. Finite Element Methods:Accuracy and Improvement [M]. Beijing:Sci-ence Press,2006.
[32]Yan Ningning. Superconvergence Analysis and a Posteriori Error Estimation in Finite Ele-ment Methods[M]. Beijing:Science Press,2008.
[33]Wang Hong. An optimal-order error estimate for an ELLAM scheme for two-dimensional linear advection-diffusion equation[J]. SIAM J. Numer. Anal.,2000,37:1338-1368.
[34]Bause M. Knabner P.. Uniform error analysis for Lagrange-Galerkin approximation of con-vection domainated problems [J]. SIAM J. Numer. Anal.,2002,39:1954-1984.
[35]Shi Dongyang, Wang Haihong, Du Yuepeng. An anisotropic noncomforming finite element for approximating a class of nonlinear Sobolev equations [J]. J. Comput. Math.,2009,27(2-3):299-314.
[36]方志朝,李宏,刘洋.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.
[37]Park C, Sheen D.W.. P1 nonconforming quadrilateral finite element methods for second-order elliptic problems [J]. SIAM J. Numer. Anal.,2003,41:624-640
[38]Lin Qun, Tobiska L.. Zhou Aihui. Superconvergence and extrapolation of nonconforming low-order elements applied to the Poisson equation [J]. IMA J. Numer. Anal.,2005,25:160-181.
[39]Rannacher R., Turek S.. Simple nonconforming quadrilateral stokes element [J]. Numer. Meth. for PDE,1992,8:97-111.
[40]胡俊,满红英,石钟慈.带约束非协调旋转Q1元在Stokes和平面弹性问题中的应用[J].计算数学,2005,27(3)：311-324.
[41]Hu Jun, Shi Zhongci. Constrained quadrilateral nonconforming rotated Q1-element[J]. J. Comput. Math.,2005,23(5):561-586.
[42]Liu Huipo, Yan Ningning. Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations [J]. Adv. Comput. Math.,2008,29:375-392.
[43]Apel T., Nicaise S., Schoberl J.. Crouzeix-Raviart type finite element on anisotropic meshes[J]. Numer. Math.,2001,89:193-293.
[44]Shi Dongyang, Zhang Buying. High accuracy analysis for the finite element method for nonlinear viscoelastic wave equations with nonlinear boundary conditions [J]. J. Syst. Sci. Complex.,2011,24:795-802.
[45]Shi Dongyang, Zhang Buying. High accuracy analysis of the finite elemnet method for nonlinear parabolic integro-differential equations with nonlinear boundary conditions[J]数学进展,2009,38(6)：715-722.
[46]石东洋,王慧敏.非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近[J].数学物理学报,2010,30A(4)：1018-1029.
[47]Knobloch P., Tobiska L.. The P1mod element:a new nonconforming finite elemnet for convection-diffusion problems[J]. SIAM J. Numer. Anal.,2003,41:436-456.
[48]石东洋,梁慧.各向异性网格下线性三角形元的超收敛性分析[J].工程数学学报，2007,24(3)：487-493.
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