强阻尼波动方程及粘弹性方程的高效有限元分析
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摘要
本论文包括两部分：第一部分主要讨论半线性强阻尼波动方程的高精度分析.首先,利用双线性元和零阶R-T元对该方程提出了一个新的具有BB条件自然满足,总体自由度最少等优势的矩形混合元逼近格式,导出了有关变量的超逼近和整体超收敛结果.其次,研究了双线性元对该方程的有限元逼近,借助于该元已有的分析估计式和插值后处理技巧,导出了H1模意义下的超逼近和超收敛结果.同时,通过构造一个新的合适的外推格式,得到了三阶外推解.第二部分主要研究粘弹性方程的H1-Galerkin扩展混合元方法,该方法能同时得到原始未知函数,梯度和流量(梯度乘以系数)三个变量在半离散和全离散格式下的最优阶误差估计,其中有限元逼近空间不要求满足BB条件,网格剖分不需要满足拟一致假设.
This paper includes two parts:In the first part, we focus on higher accuracy analysis for strongly damping wave equation. Firstly, a new rectangular mixed finite element scheme for this equation is proposed based on the bilinear element and zero-order R-T element, which has the advantages:the BB condition is satisfied automatically and the total degrees of freedom involved are lowest etc. The global superclose and superconvergence results are obtained. Secondly, the bilinear element approximation for this equation is studied, the su-perclose and superconvergence results of H1norm are derived by virtue of the known error estimates of the bilinear element and the interpolation post-processing techniques. More-over, the three-order extrapolation solution is achieved through constructing a new and suitable extrapolation scheme. In the second part, we discuss an H1-Galerkin expanded mixed finite element method for viscoelasticity type equation, the optimal approximation of three variables:the unknown function, its gradient and its flux(the gradient multiplies the coefficients) are presented in which the finite element spaces do not need to satisfy the BB condition, the mesh generation need not to be quasi-uniform assumption.
This paper includes two parts:In the first part, we focus on higher accuracy analysis for strongly damping wave equation. Firstly, a new rectangular mixed finite element scheme for this equation is proposed based on the bilinear element and zero-order R-T element, which has the advantages:the BB condition is satisfied automatically and the total degrees of freedom involved are lowest etc. The global superclose and superconvergence results are obtained. Secondly, the bilinear element approximation for this equation is studied, the su-perclose and superconvergence results of H1norm are derived by virtue of the known error estimates of the bilinear element and the interpolation post-processing techniques. More-over, the three-order extrapolation solution is achieved through constructing a new and suitable extrapolation scheme. In the second part, we discuss an H1-Galerkin expanded mixed finite element method for viscoelasticity type equation, the optimal approximation of three variables:the unknown function, its gradient and its flux(the gradient multiplies the coefficients) are presented in which the finite element spaces do not need to satisfy the BB condition, the mesh generation need not to be quasi-uniform assumption.
引文
[1]R.A. Adams. Sobolev Spaces[M]. New York:Academic Press,1975.
[2]P.G. Ciarlet. The Finite Element Method for Elliptic Problems[M]. North-Holland:Amster-dam,1978.
[3]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.
[4]罗振东.混合有限元法及其应用[M].北京：科学出版社,2006.
[5]F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods[M]. Springer-Verlag,1991.
[6]I. Babuska. Error-bounds for finite element method[J]. Numer. Math.,1971,16:322-333.
[7]F. Brezzi. On the exiatence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[J]. RAIRO Anal. Numer.,1974,8(2):129-151.
[8]R.S. Falk, J.E. Osborn. Error estimates for mixed methods[J]. RAIRO Anal. Numer.,1980, 14(3):249-277.
[9]方道元,徐江.强阻尼波方程解的整体存在性和一致衰减性[J].数学物理学报,2006,26A(5)：753-765.
[10]徐润章,刘博为.四阶具强阻尼非线性波动方程解的整体存在性与不存在性[J].数学年刊,2011,32A(3)：267-276.
[11]V. Belleri, V. Pata. Attractors for semilinear strongly damped wave equation on R3[J]. Discrete Contin. Dynam. Systems,2001,7(4):719-735.
[12]V. Pata. Attractors for a damped wave equation on R3 with linear memory [J]. Math. Meth. Appl. Sci.,2000,23:633-653.
[13]孙淑珍.四阶强阻尼非线性波动方程的整体Wk,p解[J].吉林大学自然科学学报,2000,1：18-22.
[14]徐江.一类强阻尼波方程解的存在性和爆破性[J].高校应用数学学报,2006,21A(2)：157-164.
[15]周婷,向新民.无界域上半线性强阻尼波动方程的全离散有理谱逼近[J].计算数学,2009,31(4)：335-348.
[16]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.
[17]A.K. Pani, J.Y. Yuan. Mixed finite element method for a strongly damped wave equation[J]. Numer. Meth. for PDEs,2001,17:105-119.
[18]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.
[19]石东洋,唐启立,董晓静.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.
[20]陈绍春,陈红如.二阶椭圆方程新的混合元格式[J].计算数学,2010,32(2)：213-218.
[21]李明浩.两类椭圆型方程的混合有限元方法及超收敛分析[D].郑州大学,2011.
[22]孔德兴.偏微分方程[M].高等教育出版社,2010.
[23]尚亚东.方程的初边值问题[J].应用数学学报,2000,23(3)：385-393.
[24]林群,严宁宁.高效有限元构造与分析[M].石家庄：河北大学出版社,1996.
[25]Q. Lin, S.H. Zhang, N.N. Yan. Extrapolation and defect correction for diffusion equations with boundary integral condition[J]. Acta Math. Sci.,1997,17(4):405-412.
[26]石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推[J].计算数学,2005,27(4)：369-382.
[27]石东洋,任金城.Stokes问题Q2-P1混合元外推方法[J].数学的实践与认识,2008,38(17)：66-75.
[28]G. Fairweather, Q. Lin, Y.P. Lin, J.P. Wang, S.H. Zhang. Asymptotic expansions and Richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods[J]. SIAM J. Numer. Anal.,2006,44: 1122-1149.
[29]D.Y. Shi, H.Q. Zhu. The superconvergence analysis of an anisotropic finite element[J]. J. Syst. Sci. Complex.,2005,18(4):478-487.
[30]Q. Lin, J.F. Lin. Extrapolation of the bilinear element approximation for possion equation on anisotropic meshes[J]. Numer. Meth. for PDEs,2007,23(5):960-967.
[31]M.X. Li, Q. Lin. Superconvergence of finite element method of signorini problem[J]. J. Comput. Appl. Math.,2008,222(2):284-292.
[32]Q. Lin, S.H. Zhang, N.N. Yan. Asymptotic error expansion and defect correction for Sobolev and Viscoelasticity type equations [J]. J. Comput. Math.,1998,16(1):51-62.
[33]史艳华,石东洋.粘弹性方程ACM有限元的超收敛分析和外推[J].应用数学,2009,22(3)：534-541.
[34]王立俊.粘弹性方程的变网格有限元方法[J].计算数学,1990,12(2)：119-128.
[35]石东洋,关宏波.粘弹性方程的非协调变网格有限元方法[J].高校应用数学学报.2008,23(4)：452-458.
[36]D.Y. Shi, B.Y. Zhang. High accuracy analysis of the finite element method for nonlinear viscoelastic wave equation with nonlinear boundary conditions[J]. J. Syst. Sci. Complex., 2011,24(4):795-802.
[37]牛欲琪,张学凌,石东洋.粘弹性方程混合有限元超收敛分析[J].数学的实践与认识,2009,39(14)：155-162.
[38]L.P. Gao, D. Liang, B. Zhang. Error estimates for mixed finite element approximations of the viscoelasticity wave equation[J]. Math. Meth. Appl. Sci.,2004,27(17):1997-2016.
[39]H.Z. Chen, H. Wang. An optimal-order error estimate on an H1-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow[J]. Numer. Meth. for PDEs,2010, 26:188-205.
[40]Y. Liu, H. Li. A new mixed finite element method for pseudo-hyperbolic equation[J]. Math. Appl.,2010,23(1):150-157.
[41]Z.X. Chen. Expanded mixed finite element methods for linear second order elliptic problems I[J]. RAIRO Model. Math. Anal. Numer.,1998,32(4):479-499.
[42]A.K. Pani. An H1-Galerkin mixed finite element methods for parabolic partial differential equations[J]. SIAM J. Numer. Anal.,1998,35:712-727.
[43]Y. Liu, H. Li. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl. Math. Comput.,2009,212:446-457.
[44]王瑞文.双曲型积分微分方程H1-Galerkin混合元法的误差估计[J].计算数学,2006,28(1)：19-30.
[45]D.Y. Shi, Y.D. Zhang. An H1-Galerkin nonconforming mixed finite element method for pseudo-hyperbolic equations [J]. Math. Appl.,2011,24(3):448-455.
[46]陈红斌,刘晓奇.粘弹性双曲型方程的H1-Galerkin混合有限元方法[J].高等学校计算数学学报,2011,33(3)：279-288.
[47]韩小森,王明新.带非线性阻尼的粘弹性方程解的整体存在性和一致衰减性[J].数学年刊,2009,30A(1)：31-42.
[48]G.F. Webb. Existence and asymptotic behavior for a strongly damped nonlinear wave equa-tion[J]. Canad. J. Math.,1980,32(3):631-643.
[49]A.K. Pani, R.K. Sinha, A.K. Otta. An H1-Galerkin mixed method for second order hyper-bolic equations[J]. Int. J. Numer. Anal. Model.,2004,2(1):111-129.
[50]D. Zwillinger. Handbook of Differential Equation,3rd Ed[M]. Boston:Academic Press,1997.
[51]Z.X. Chen. Finite Element Methods and Their Applications[M]. Berlin:Springer-Verlag, 2005.
[52]M.F. Wheeler. A priori L2-error estimates for Galerkin approximations to parabolic differ-ential equation[J]. SIAM J. Numer. Anal.,1973,10:723-749.
[2]P.G. Ciarlet. The Finite Element Method for Elliptic Problems[M]. North-Holland:Amster-dam,1978.
[3]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.
[4]罗振东.混合有限元法及其应用[M].北京：科学出版社,2006.
[5]F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods[M]. Springer-Verlag,1991.
[6]I. Babuska. Error-bounds for finite element method[J]. Numer. Math.,1971,16:322-333.
[7]F. Brezzi. On the exiatence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[J]. RAIRO Anal. Numer.,1974,8(2):129-151.
[8]R.S. Falk, J.E. Osborn. Error estimates for mixed methods[J]. RAIRO Anal. Numer.,1980, 14(3):249-277.
[9]方道元,徐江.强阻尼波方程解的整体存在性和一致衰减性[J].数学物理学报,2006,26A(5)：753-765.
[10]徐润章,刘博为.四阶具强阻尼非线性波动方程解的整体存在性与不存在性[J].数学年刊,2011,32A(3)：267-276.
[11]V. Belleri, V. Pata. Attractors for semilinear strongly damped wave equation on R3[J]. Discrete Contin. Dynam. Systems,2001,7(4):719-735.
[12]V. Pata. Attractors for a damped wave equation on R3 with linear memory [J]. Math. Meth. Appl. Sci.,2000,23:633-653.
[13]孙淑珍.四阶强阻尼非线性波动方程的整体Wk,p解[J].吉林大学自然科学学报,2000,1：18-22.
[14]徐江.一类强阻尼波方程解的存在性和爆破性[J].高校应用数学学报,2006,21A(2)：157-164.
[15]周婷,向新民.无界域上半线性强阻尼波动方程的全离散有理谱逼近[J].计算数学,2009,31(4)：335-348.
[16]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.
[17]A.K. Pani, J.Y. Yuan. Mixed finite element method for a strongly damped wave equation[J]. Numer. Meth. for PDEs,2001,17:105-119.
[18]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.
[19]石东洋,唐启立,董晓静.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.
[20]陈绍春,陈红如.二阶椭圆方程新的混合元格式[J].计算数学,2010,32(2)：213-218.
[21]李明浩.两类椭圆型方程的混合有限元方法及超收敛分析[D].郑州大学,2011.
[22]孔德兴.偏微分方程[M].高等教育出版社,2010.
[23]尚亚东.方程的初边值问题[J].应用数学学报,2000,23(3)：385-393.
[24]林群,严宁宁.高效有限元构造与分析[M].石家庄：河北大学出版社,1996.
[25]Q. Lin, S.H. Zhang, N.N. Yan. Extrapolation and defect correction for diffusion equations with boundary integral condition[J]. Acta Math. Sci.,1997,17(4):405-412.
[26]石东洋,梁慧.一个新的非常规Hermite型各向异性矩形元的超收敛分析及外推[J].计算数学,2005,27(4)：369-382.
[27]石东洋,任金城.Stokes问题Q2-P1混合元外推方法[J].数学的实践与认识,2008,38(17)：66-75.
[28]G. Fairweather, Q. Lin, Y.P. Lin, J.P. Wang, S.H. Zhang. Asymptotic expansions and Richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods[J]. SIAM J. Numer. Anal.,2006,44: 1122-1149.
[29]D.Y. Shi, H.Q. Zhu. The superconvergence analysis of an anisotropic finite element[J]. J. Syst. Sci. Complex.,2005,18(4):478-487.
[30]Q. Lin, J.F. Lin. Extrapolation of the bilinear element approximation for possion equation on anisotropic meshes[J]. Numer. Meth. for PDEs,2007,23(5):960-967.
[31]M.X. Li, Q. Lin. Superconvergence of finite element method of signorini problem[J]. J. Comput. Appl. Math.,2008,222(2):284-292.
[32]Q. Lin, S.H. Zhang, N.N. Yan. Asymptotic error expansion and defect correction for Sobolev and Viscoelasticity type equations [J]. J. Comput. Math.,1998,16(1):51-62.
[33]史艳华,石东洋.粘弹性方程ACM有限元的超收敛分析和外推[J].应用数学,2009,22(3)：534-541.
[34]王立俊.粘弹性方程的变网格有限元方法[J].计算数学,1990,12(2)：119-128.
[35]石东洋,关宏波.粘弹性方程的非协调变网格有限元方法[J].高校应用数学学报.2008,23(4)：452-458.
[36]D.Y. Shi, B.Y. Zhang. High accuracy analysis of the finite element method for nonlinear viscoelastic wave equation with nonlinear boundary conditions[J]. J. Syst. Sci. Complex., 2011,24(4):795-802.
[37]牛欲琪,张学凌,石东洋.粘弹性方程混合有限元超收敛分析[J].数学的实践与认识,2009,39(14)：155-162.
[38]L.P. Gao, D. Liang, B. Zhang. Error estimates for mixed finite element approximations of the viscoelasticity wave equation[J]. Math. Meth. Appl. Sci.,2004,27(17):1997-2016.
[39]H.Z. Chen, H. Wang. An optimal-order error estimate on an H1-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow[J]. Numer. Meth. for PDEs,2010, 26:188-205.
[40]Y. Liu, H. Li. A new mixed finite element method for pseudo-hyperbolic equation[J]. Math. Appl.,2010,23(1):150-157.
[41]Z.X. Chen. Expanded mixed finite element methods for linear second order elliptic problems I[J]. RAIRO Model. Math. Anal. Numer.,1998,32(4):479-499.
[42]A.K. Pani. An H1-Galerkin mixed finite element methods for parabolic partial differential equations[J]. SIAM J. Numer. Anal.,1998,35:712-727.
[43]Y. Liu, H. Li. H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations[J]. Appl. Math. Comput.,2009,212:446-457.
[44]王瑞文.双曲型积分微分方程H1-Galerkin混合元法的误差估计[J].计算数学,2006,28(1)：19-30.
[45]D.Y. Shi, Y.D. Zhang. An H1-Galerkin nonconforming mixed finite element method for pseudo-hyperbolic equations [J]. Math. Appl.,2011,24(3):448-455.
[46]陈红斌,刘晓奇.粘弹性双曲型方程的H1-Galerkin混合有限元方法[J].高等学校计算数学学报,2011,33(3)：279-288.
[47]韩小森,王明新.带非线性阻尼的粘弹性方程解的整体存在性和一致衰减性[J].数学年刊,2009,30A(1)：31-42.
[48]G.F. Webb. Existence and asymptotic behavior for a strongly damped nonlinear wave equa-tion[J]. Canad. J. Math.,1980,32(3):631-643.
[49]A.K. Pani, R.K. Sinha, A.K. Otta. An H1-Galerkin mixed method for second order hyper-bolic equations[J]. Int. J. Numer. Anal. Model.,2004,2(1):111-129.
[50]D. Zwillinger. Handbook of Differential Equation,3rd Ed[M]. Boston:Academic Press,1997.
[51]Z.X. Chen. Finite Element Methods and Their Applications[M]. Berlin:Springer-Verlag, 2005.
[52]M.F. Wheeler. A priori L2-error estimates for Galerkin approximations to parabolic differ-ential equation[J]. SIAM J. Numer. Anal.,1973,10:723-749.