两类发展方程的数值方法
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摘要
本文讨论了两类发展方程-Sobolev方程初边值问题和均匀棒纯纵向运动方程初边值问题的数值方法，得到了这两类问题离散格式的误差估计。
第一章讨论Sobolev方程初边值问题
的扩展混合元方法。若采用标准有限元方法，对解空间的光滑度相对要求较高并且在进行误差估计时，只能直接得到关于未知纯量的误差估计。采用传统的混合有限元方法，不但降低了对解空间光滑度的要求，而且还可以同时高精度的对未知纯量及流量进行估计。该方法是传统混合元方法的一种推广，它能同时逼近未知函数、梯度、流量，较好地刻画了具有混合边界条件的Sobolev方程初边值问题，同时避免了对小系数进行求逆。数值分析结果说明扩展混合元方法是稳定的，得到了逼近以上三个量的最优L~2误差估计和关于未知函数u的拟最优的L~∞估计。
第二章讨论均匀棒纯纵向运动初边值问题
的有限元方法。这是引起广泛关注的一类重要的非线性发展方程，它典型反映了一类自由应力状态下均匀粘弹性棒的纯纵向运动问题。对于此问题的研究仅限于差分
方法及稳定性估计等，本文则给出了有限元方法半离散格式和全离散格式的误差分
析，得到了离散解逼近未知函数u的关于空间和时间的最优尸误差估计.最后我
们通过一个数值例子来进一步说明我们得到的误差估计是合理的.
In this paper , we consider several neumrical methods for the initial-boundary value problems of Sobolev equations and purely longtudinal motion equations of a homogeneous bar, obtain the error estimates of the discrete schemes for the two kinds of problems.
In Chapter one,we consider the Expanded Mixed finite element methods for the Sobolev equations
This method expands the standard mixed formulation in the sense that three variable are explixitly treatedithe scalar unknwon, its gradient and its flux. Based on this fomulation,expanded mixed finite element approximations of the Sobolev equations are considered.Optimal order error estimates for the scalar unknwon, its gradient and its flux in L2-norms are obtained for this new mixed formulation.Also, Quasi-optimal order estimates are obtained for the approximations of the the scalar unknwon.
In Chapter two, we consider the finite element methods for the following initial-
value problem of purely longtudinal motion of a homogeneous bar
In this chapter,we give the error analysis of this discrete schemes and get op timal error estimates for the discrete solution of u.
第一章讨论Sobolev方程初边值问题
的扩展混合元方法。若采用标准有限元方法，对解空间的光滑度相对要求较高并且在进行误差估计时，只能直接得到关于未知纯量的误差估计。采用传统的混合有限元方法，不但降低了对解空间光滑度的要求，而且还可以同时高精度的对未知纯量及流量进行估计。该方法是传统混合元方法的一种推广，它能同时逼近未知函数、梯度、流量，较好地刻画了具有混合边界条件的Sobolev方程初边值问题，同时避免了对小系数进行求逆。数值分析结果说明扩展混合元方法是稳定的，得到了逼近以上三个量的最优L~2误差估计和关于未知函数u的拟最优的L~∞估计。
第二章讨论均匀棒纯纵向运动初边值问题
的有限元方法。这是引起广泛关注的一类重要的非线性发展方程，它典型反映了一类自由应力状态下均匀粘弹性棒的纯纵向运动问题。对于此问题的研究仅限于差分
方法及稳定性估计等，本文则给出了有限元方法半离散格式和全离散格式的误差分
析，得到了离散解逼近未知函数u的关于空间和时间的最优尸误差估计.最后我
们通过一个数值例子来进一步说明我们得到的误差估计是合理的.
In this paper , we consider several neumrical methods for the initial-boundary value problems of Sobolev equations and purely longtudinal motion equations of a homogeneous bar, obtain the error estimates of the discrete schemes for the two kinds of problems.
In Chapter one,we consider the Expanded Mixed finite element methods for the Sobolev equations
This method expands the standard mixed formulation in the sense that three variable are explixitly treatedithe scalar unknwon, its gradient and its flux. Based on this fomulation,expanded mixed finite element approximations of the Sobolev equations are considered.Optimal order error estimates for the scalar unknwon, its gradient and its flux in L2-norms are obtained for this new mixed formulation.Also, Quasi-optimal order estimates are obtained for the approximations of the the scalar unknwon.
In Chapter two, we consider the finite element methods for the following initial-
value problem of purely longtudinal motion of a homogeneous bar
In this chapter,we give the error analysis of this discrete schemes and get op timal error estimates for the discrete solution of u.
引文
[1] R.E.Ewing, The approximation of certain parabolic equation backward in time by Sobolev equations, SIAM Numer. Anal,23(1975),283-294.
[2] L.C.Cowsar, T.F.Dupont, and M.F.Wheeler, Apriori estimates for mixed finite element approximations of second order hyperbolic equations with absorbying boundary conditions, SIAM J.Numer.Anal.,33(1996),492-504.
[3] M.Celia and P.Binning, Two-phase unsaturated flow one dimensional simulation and air phase velocities. Water Resouces Research,28(1992),2819-2828.
[4] J.Touma and M.Vauclin, Experimental and numrical analysis of two-phase infiltration in a partially saturated soil, Transport in Porious Media 1(1986),27-55.
[5] F.Brezzi, J.Douglas.Jr, M.Fortin and L.Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modèl. Math.Anal. Numér,21(1987),581-604.
[6] F.Brezzi, J.Douglas.Jr, R.Durán and M.Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer.Math,51(1987),237-250.
[7] F.Brezzi, J.Douglas.Jr. and L.Marini, Two families of mixed finite elements for second order elliptic problems, Numer.Math,47(1985),217-235.
[8] Z.Chen and J.Douglas.Jr, Prismatic mixed finite elements for second order elliptic problems,Calcolo, 26(1989).
[9] J.C.Nedelec, Mixed finite elements in R~3, Numer.Math,35(1980),315-341.
[10] J.C.Nedelec, A new family of mixed finite elements in R~3, Numer. Math, 50(1986),57-81.
[11] P.A.Raviart and J.M.Thomas, A mixed finite element method for second order elliptic problems, Lecture Notes in Math. 606,Springer,Brelin,1977,pp.292-315.
[12] Zhangxin Chen, Expanded mixed finite element methods for linear second-order elliptic problems. Numer.Math,vol.32,no.4(1998),479-499.
[13] Z.Chen, Expanded mixed finite element methods,M~2AN,vol.32.n~04,1998,503-520.
[14] J.Wang, Asymptotic expansions and L~∞-error estimates for mixed finite element methods for second order elliptic problems,Numer.Math.,55(1989),401-430.
[15] G.Andrews, On the existence of solution to the Equation u_(tt)= u_(xxt)+σ(u_x)_x, J.Difference Equation,35(1980),200-231.
[16] 刘亚成，刘大成，方程u_(tt)=u_(xxt)+σ(u_x)_x的初边值问题，周期边界问题与初值问题，数学年刊，9A：4(1988)，459-470．
[17] 高兴宝，万桂华，陈开周，方程u_(tt)-u_(xxt)+σ(u_x)_x的初边值问题的差分法．计算数学，5(2000)，167-176．
[18] P.G.Ciarlet, The finite element methods for elliptic problems, North-Holland.1978.
[19] 罗振东．有限元混合法理论基础及其应用-发展与应用．济南，山东教育出版社，1996．
[20] 罗振东，刘儒勋．Bergers方程的混合元分析及其数值模拟，计算数学，3(1999)，257-267．
[21] 罗振东，非常的热传导对流问题的混合有限元方法，计算数学，1998，20(1)，69-88．
[22] 李潜，可压核废料污染问题的有限元方法，计算数学，16：3(1994)，227-232．
[23] 陈焕祯，不可压混溶驱动问题的流线扩散混合元数值模拟，系统科学与数学，18(1)，1998，87-95．
[24] 陈焕祯，李潜，二维不可压无粘流动问题沿流线的有限元数值模拟，工程数学学报，13(1)，1996．
[25] M.Peter, A Mixed Finite Element Method For The Biarmonic Equation, SIAM J. NUMER. ANAL. 1987.24(4), 737-749.
[26] J.Doulags Jr. and J.Wang, A new family of mixed finite elment spaces over tectangles, Mat.Aplic.Couput. 12(1993). 183-193.
[27] 姜子文，L~∞(L2)and L~∞(L~∞) Error estimates for mixed methods for integro-differential equations of parabolic type, Math. Model. Numer. Anal., V. 33, N. 3 (1999), 531-546.
[28] 姜子文，李潜，非线性双曲型方程的插值全离散有限元方法的超收敛性，工程数学学报，V．16，N．2(1999)，1-8．
[29] 陈焕祯，姜子文，L~∞-convergence of mixed finite element method for Laplacian operator, Korea J. Computation and Applied Mathematics, 12. V. 7, N. 1, (2000), 13. 61-83.
[30] 袁益让，王宏，非线性双曲型方程有限元方法的误差估计，系统科学与数学，1985，5(3)，161-171．
[31] 杨青，二位热传导型半导体问题的一个二阶格式，数学物理学报，2001，21(增刊)，683-692．
[32] 刘中艳，两类方程混合元方法的数值模拟．硕士毕业论文，2003．
[33] M.Peter, A Mixed Finite Element Method For The Biarmonic Equation, SIAM J.NUMER.ANAL.1987.24(4),737-749.
[34] J.Doulags Jr. and J.Wang, A new family of mixed finite elment spaces over tectangles, Mat.Aplic.Couput. 12(1993). 183-193.
[2] L.C.Cowsar, T.F.Dupont, and M.F.Wheeler, Apriori estimates for mixed finite element approximations of second order hyperbolic equations with absorbying boundary conditions, SIAM J.Numer.Anal.,33(1996),492-504.
[3] M.Celia and P.Binning, Two-phase unsaturated flow one dimensional simulation and air phase velocities. Water Resouces Research,28(1992),2819-2828.
[4] J.Touma and M.Vauclin, Experimental and numrical analysis of two-phase infiltration in a partially saturated soil, Transport in Porious Media 1(1986),27-55.
[5] F.Brezzi, J.Douglas.Jr, M.Fortin and L.Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modèl. Math.Anal. Numér,21(1987),581-604.
[6] F.Brezzi, J.Douglas.Jr, R.Durán and M.Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer.Math,51(1987),237-250.
[7] F.Brezzi, J.Douglas.Jr. and L.Marini, Two families of mixed finite elements for second order elliptic problems, Numer.Math,47(1985),217-235.
[8] Z.Chen and J.Douglas.Jr, Prismatic mixed finite elements for second order elliptic problems,Calcolo, 26(1989).
[9] J.C.Nedelec, Mixed finite elements in R~3, Numer.Math,35(1980),315-341.
[10] J.C.Nedelec, A new family of mixed finite elements in R~3, Numer. Math, 50(1986),57-81.
[11] P.A.Raviart and J.M.Thomas, A mixed finite element method for second order elliptic problems, Lecture Notes in Math. 606,Springer,Brelin,1977,pp.292-315.
[12] Zhangxin Chen, Expanded mixed finite element methods for linear second-order elliptic problems. Numer.Math,vol.32,no.4(1998),479-499.
[13] Z.Chen, Expanded mixed finite element methods,M~2AN,vol.32.n~04,1998,503-520.
[14] J.Wang, Asymptotic expansions and L~∞-error estimates for mixed finite element methods for second order elliptic problems,Numer.Math.,55(1989),401-430.
[15] G.Andrews, On the existence of solution to the Equation u_(tt)= u_(xxt)+σ(u_x)_x, J.Difference Equation,35(1980),200-231.
[16] 刘亚成，刘大成，方程u_(tt)=u_(xxt)+σ(u_x)_x的初边值问题，周期边界问题与初值问题，数学年刊，9A：4(1988)，459-470．
[17] 高兴宝，万桂华，陈开周，方程u_(tt)-u_(xxt)+σ(u_x)_x的初边值问题的差分法．计算数学，5(2000)，167-176．
[18] P.G.Ciarlet, The finite element methods for elliptic problems, North-Holland.1978.
[19] 罗振东．有限元混合法理论基础及其应用-发展与应用．济南，山东教育出版社，1996．
[20] 罗振东，刘儒勋．Bergers方程的混合元分析及其数值模拟，计算数学，3(1999)，257-267．
[21] 罗振东，非常的热传导对流问题的混合有限元方法，计算数学，1998，20(1)，69-88．
[22] 李潜，可压核废料污染问题的有限元方法，计算数学，16：3(1994)，227-232．
[23] 陈焕祯，不可压混溶驱动问题的流线扩散混合元数值模拟，系统科学与数学，18(1)，1998，87-95．
[24] 陈焕祯，李潜，二维不可压无粘流动问题沿流线的有限元数值模拟，工程数学学报，13(1)，1996．
[25] M.Peter, A Mixed Finite Element Method For The Biarmonic Equation, SIAM J. NUMER. ANAL. 1987.24(4), 737-749.
[26] J.Doulags Jr. and J.Wang, A new family of mixed finite elment spaces over tectangles, Mat.Aplic.Couput. 12(1993). 183-193.
[27] 姜子文，L~∞(L2)and L~∞(L~∞) Error estimates for mixed methods for integro-differential equations of parabolic type, Math. Model. Numer. Anal., V. 33, N. 3 (1999), 531-546.
[28] 姜子文，李潜，非线性双曲型方程的插值全离散有限元方法的超收敛性，工程数学学报，V．16，N．2(1999)，1-8．
[29] 陈焕祯，姜子文，L~∞-convergence of mixed finite element method for Laplacian operator, Korea J. Computation and Applied Mathematics, 12. V. 7, N. 1, (2000), 13. 61-83.
[30] 袁益让，王宏，非线性双曲型方程有限元方法的误差估计，系统科学与数学，1985，5(3)，161-171．
[31] 杨青，二位热传导型半导体问题的一个二阶格式，数学物理学报，2001，21(增刊)，683-692．
[32] 刘中艳，两类方程混合元方法的数值模拟．硕士毕业论文，2003．
[33] M.Peter, A Mixed Finite Element Method For The Biarmonic Equation, SIAM J.NUMER.ANAL.1987.24(4),737-749.
[34] J.Doulags Jr. and J.Wang, A new family of mixed finite elment spaces over tectangles, Mat.Aplic.Couput. 12(1993). 183-193.