Sobolev方程的间断时间变量的Galerkin有限元方法
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摘要
Sobolev方程在流体力学、热力学等许多数学物理方面都有着广泛的应用,例如:流体穿过裂缝岩石的渗透理论,土壤中的湿气迁移问题,不同介质间的热传导问题等等.本文共分两章:
第一章介绍了拟线性Sobolev方程的间断时间变量的Galerkin有限元方法.
有限元法对全离散格式的推导,常见的是:先用Galerkin有限元法对空间变量进行离散化,再用有限差分对时间变量进行离散化.时间间断的Galerkin有限元法是对时间变量也应用Galerkin方法进行离散化,构造关于时间变量是次数不超过q-1次的分片多项式形式的有限元格式.这样在方法的定义和分析上,对时间变量的处理与对空间变量的处理是类似的,从而对时间变量和空间变量都可以获得高精度.
本章研究了拟线性Sobolev方程的间断时间变量的Galerkin有限元方法.按照上述的基本思路,构造了关于时间变量是次数不超过q-1的分片多项式形式的有限元格式,并进行了误差分析,得到了最优的L~∞(O,T;H~1(Ω))模误差估计.拟线性Sobolev方程含有时间和空间的混合偏导数项u_(xxt),从而增加了研究的难度,也是本文研究的意义所在.
本章分为五节:第一节是引言,简单介绍了间断时间变量的Galerkin有限元方法;第二节和第三节给出了拟线性Sobolev方程的模型以及关于时间间断的Galerkin有限元格式;第四节通过不动点定理证明了格式解的存在唯一性;第五节通过引入非标准椭圆投影,借助反向问题等方法,对该格式进行了L~∞(O,T;H~1(Ω))模误差估计.
第二章介绍了线性Sobolev方程的间断时间变量的H~1-Galerkin混合有限元方法.
H~1-Galerkin混合有限元方法首先将问题化成未知变量u和其通量函数σ的一阶混合方程组,然后用H~1-Galerkin有限元方法离散.在这个方法中,未知变量u和其通量流函数σ的逼近空间V_h×W_h,可以选择不同次数的多项式空间.与标准混合元法相比,H~1-Galerkin混合元方法不需要满足空间的LBB-相容条件,误差估计可以很好地区分出V_h和W_h的逼近效果,而且没有要求有限元网格的拟一致性条件.
本章利用H~1-Galerkin混合有限元方法和间断时空有限元方法相结合的技巧,研究了线性Sobolev方程的间断时间变量的H~1-Galerkin混合有限元方法.构造了关于时间变量是次数不超过q-1的分片多项式形式的H~1-Galerkin有限元格式,发挥两个方法的优势,获得了高精度数值方法.
本章也分为五节:第一节是前言,简单介绍了H~1-Galerkin混合有限元方法;第二节和第三节分别给出了线性Sobolev方程的模型以及关于时间间断的H~1-Galerkin混合有限元格式;第四节通过Gronwall不等式证明了格式解的存在唯一性;第五节进行了误差分析,得到了未知函数u和通量函数σ的最优的H~1模误差估计,对空间变量和时间变量的逼近阶都达到了最优.
There has been a wide range of applications of Sobolev equations in fluid mechanics, thermodynamics,and many other aspects of mathematical physics,such as the percolation theory when the fluid fllows through the cracks,the transfer problem of the moisture in the soil,and the heat conduction problem in different materials and so on.This paper is divided into two chapters.
Chapter 1 introduces the time discontinuous Galerkin finite element method for the quasilinear Sobolev equations.
The completely discrete schemes are derived by diseretizing in the space variables by means of a Galerkin finite element method and then discretizing the time variable by a finite difference method.The time discontinuous Galerkin finite element method is to discrete the time variable also by the Galerkin method,and construct the scheme which treats the time and sapce variables similarly.The approximate solution will be sought as a piecewise polynomial of degree at most q-1 about time variable.Then,we can get high accuracy for both the space variable and the time variable.
According to the above methods,this chapter built the time discontinuous Galerkin finite element scheme for the quasilinear Sobolev equations.The approximate solution was sought as a piecewise polynomial of degree at most q-1 about time variable.The error estimates were derived.The quasilinar Sobolev equations has the mixed differential term u_(xxt),which increases the difficulty for the study,and it is also the significance of this paper.
This chapter is divided into five sections.SectionⅠintroduces the time dis- continuous Galerkin finite element method briefly.In sectionsⅡandⅢ,we give the mathematical model and the time discontinuous Galerkin finite element scheme. respectively.In sectionⅣ,the existence and uniqueness of the solution are proved by the fixed-point theorem.In sectionⅤ,we make the L~∞(0,T;H~1(Ω))-norm error estimates by a non-standard elliptic projection and a inverse problem.
Chapetr 2 introduces the time diseontinnous H~1-Galerkin mixed finite element method for the linear Sobolev equations.
First,H~1-Galerkin mixed finite element method changes the model into a first-order system about unknown variable u and its fluxσ,then use H~1-Galerkin finite element method to it.The approximating finite element spaces V_h×W_h,to variable u and fluxσrespectively,were allowed to be of different polynomial degrees.Hence, estimations can be obtained which distinguished the better approximation properties of V_h and W_h.Compared to standard mixed methods,the H~1-Galerkin mixed finite element method was not subject to LBB-consistency condition.Moreover,the quasiuniformity condition was not imposed on the finite element mesh.
To get high precision numerical method,this chapter combined the advantages of H~1-Galerkin mixed finite element method and the time discontinuous Galerkin finite element method to linear Sobolev equations.The schemes were bulit to approximate solution by a piecewise polynomial of degree at most q-1 about time variable.The optimal error estimates were derived.
This chapter is also divided into five sections.SectionⅠintroduces the H~1-Galerkin mixed finite element method briefly.In sectionsⅡandⅢ,we give the mathematical model of the linear Sobolev equations and the time discontinuous H~1-Galerkin mixed finite element scheme,respectively.In sectionⅣ,the existence and uniqueness of the solution are proved by the Gronwall inequality.In sectionⅤ,we make the error analysis and achieve H~1-norm error estimates for variable u and its fluxσ,where the convergence orders are optimal both about the space variable and the time variable.
第一章介绍了拟线性Sobolev方程的间断时间变量的Galerkin有限元方法.
有限元法对全离散格式的推导,常见的是:先用Galerkin有限元法对空间变量进行离散化,再用有限差分对时间变量进行离散化.时间间断的Galerkin有限元法是对时间变量也应用Galerkin方法进行离散化,构造关于时间变量是次数不超过q-1次的分片多项式形式的有限元格式.这样在方法的定义和分析上,对时间变量的处理与对空间变量的处理是类似的,从而对时间变量和空间变量都可以获得高精度.
本章研究了拟线性Sobolev方程的间断时间变量的Galerkin有限元方法.按照上述的基本思路,构造了关于时间变量是次数不超过q-1的分片多项式形式的有限元格式,并进行了误差分析,得到了最优的L~∞(O,T;H~1(Ω))模误差估计.拟线性Sobolev方程含有时间和空间的混合偏导数项u_(xxt),从而增加了研究的难度,也是本文研究的意义所在.
本章分为五节:第一节是引言,简单介绍了间断时间变量的Galerkin有限元方法;第二节和第三节给出了拟线性Sobolev方程的模型以及关于时间间断的Galerkin有限元格式;第四节通过不动点定理证明了格式解的存在唯一性;第五节通过引入非标准椭圆投影,借助反向问题等方法,对该格式进行了L~∞(O,T;H~1(Ω))模误差估计.
第二章介绍了线性Sobolev方程的间断时间变量的H~1-Galerkin混合有限元方法.
H~1-Galerkin混合有限元方法首先将问题化成未知变量u和其通量函数σ的一阶混合方程组,然后用H~1-Galerkin有限元方法离散.在这个方法中,未知变量u和其通量流函数σ的逼近空间V_h×W_h,可以选择不同次数的多项式空间.与标准混合元法相比,H~1-Galerkin混合元方法不需要满足空间的LBB-相容条件,误差估计可以很好地区分出V_h和W_h的逼近效果,而且没有要求有限元网格的拟一致性条件.
本章利用H~1-Galerkin混合有限元方法和间断时空有限元方法相结合的技巧,研究了线性Sobolev方程的间断时间变量的H~1-Galerkin混合有限元方法.构造了关于时间变量是次数不超过q-1的分片多项式形式的H~1-Galerkin有限元格式,发挥两个方法的优势,获得了高精度数值方法.
本章也分为五节:第一节是前言,简单介绍了H~1-Galerkin混合有限元方法;第二节和第三节分别给出了线性Sobolev方程的模型以及关于时间间断的H~1-Galerkin混合有限元格式;第四节通过Gronwall不等式证明了格式解的存在唯一性;第五节进行了误差分析,得到了未知函数u和通量函数σ的最优的H~1模误差估计,对空间变量和时间变量的逼近阶都达到了最优.
There has been a wide range of applications of Sobolev equations in fluid mechanics, thermodynamics,and many other aspects of mathematical physics,such as the percolation theory when the fluid fllows through the cracks,the transfer problem of the moisture in the soil,and the heat conduction problem in different materials and so on.This paper is divided into two chapters.
Chapter 1 introduces the time discontinuous Galerkin finite element method for the quasilinear Sobolev equations.
The completely discrete schemes are derived by diseretizing in the space variables by means of a Galerkin finite element method and then discretizing the time variable by a finite difference method.The time discontinuous Galerkin finite element method is to discrete the time variable also by the Galerkin method,and construct the scheme which treats the time and sapce variables similarly.The approximate solution will be sought as a piecewise polynomial of degree at most q-1 about time variable.Then,we can get high accuracy for both the space variable and the time variable.
According to the above methods,this chapter built the time discontinuous Galerkin finite element scheme for the quasilinear Sobolev equations.The approximate solution was sought as a piecewise polynomial of degree at most q-1 about time variable.The error estimates were derived.The quasilinar Sobolev equations has the mixed differential term u_(xxt),which increases the difficulty for the study,and it is also the significance of this paper.
This chapter is divided into five sections.SectionⅠintroduces the time dis- continuous Galerkin finite element method briefly.In sectionsⅡandⅢ,we give the mathematical model and the time discontinuous Galerkin finite element scheme. respectively.In sectionⅣ,the existence and uniqueness of the solution are proved by the fixed-point theorem.In sectionⅤ,we make the L~∞(0,T;H~1(Ω))-norm error estimates by a non-standard elliptic projection and a inverse problem.
Chapetr 2 introduces the time diseontinnous H~1-Galerkin mixed finite element method for the linear Sobolev equations.
First,H~1-Galerkin mixed finite element method changes the model into a first-order system about unknown variable u and its fluxσ,then use H~1-Galerkin finite element method to it.The approximating finite element spaces V_h×W_h,to variable u and fluxσrespectively,were allowed to be of different polynomial degrees.Hence, estimations can be obtained which distinguished the better approximation properties of V_h and W_h.Compared to standard mixed methods,the H~1-Galerkin mixed finite element method was not subject to LBB-consistency condition.Moreover,the quasiuniformity condition was not imposed on the finite element mesh.
To get high precision numerical method,this chapter combined the advantages of H~1-Galerkin mixed finite element method and the time discontinuous Galerkin finite element method to linear Sobolev equations.The schemes were bulit to approximate solution by a piecewise polynomial of degree at most q-1 about time variable.The optimal error estimates were derived.
This chapter is also divided into five sections.SectionⅠintroduces the H~1-Galerkin mixed finite element method briefly.In sectionsⅡandⅢ,we give the mathematical model of the linear Sobolev equations and the time discontinuous H~1-Galerkin mixed finite element scheme,respectively.In sectionⅣ,the existence and uniqueness of the solution are proved by the Gronwall inequality.In sectionⅤ,we make the error analysis and achieve H~1-norm error estimates for variable u and its fluxσ,where the convergence orders are optimal both about the space variable and the time variable.
引文
[1]Vidar rrhomee著,黄明游等译,抛物问题有限元法,吉林大学出版社,长春,1986.
[2]M.Deflour,W.Hager and F.Trochu,Discontinuous Galerkin methods for ordinary differential equations,Math.Comput.36(1981) 455-473.
[3]P.Lcsaint and P.Ravarit,On a finite element method for solving the neutron transport equation,Mathematical Aspects of Finite Elements in Partial Differential Equations,C.de Boor,ed.,Academic Press,New York,1974,89-123.
[4]P.Jamet,Galerkin-type approxiamtions which are discontinuous in time for parabolic equations in a variable domain,SIAM J,Numer.Anal.,15(1978)912-928.
[5]K.Eriksson,C.,Johnson and V.Thomee,Time discretization of parabolic problems by the discontinuous Galerkin method,RAIRO Math.Anal.,19(1985)611-643.
[6]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems I:A linear model problems,SIAM,J,Numer.Anal.,28(1991) 43-77.
[7]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅱ:Optimal error estimates in L_∞L_2 and L_∞L_∞,SIAM J,Numer.Anal.,32(1995) 706-740.
[8]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅳ:Nonlinear problems,SIAM J,Numer.Anal.,32(1995) 1729-1749.
[9]CH.G.Makridakis and I.Babuska,On the stability of the discontinuous Galerkin method for the heat equation,SIAM J,Numer.Anal.,34(1997) 389-401.
[10]G.I.Barenblatt,I.P.Zheltov and I.N.Kochina,Basic concepts in the theory of seepage of homogenous liquids in fissured rocks,J,Appl.Math.Mech.,24(1960) 1286-1303.
[11]施德明,非线性湿气迁移方程的初边值问题,应用数学学报,13(1990)31-38.
[12]Ting T W,A cooling process according to two-temperature theory of heat conduction.J Math Anal Appl,45(1974) 23-31.
[13]P.J.Chen and M.E.Gurtin,On a theory of heat conduction involving two temperatures,Z.Angew.Math.Phys.,19(1968) 614-627.
[14]P.L.Davis,A quasilinear parabolic and a related third order problem,J.Math.Anal.Appl.,49(1970) 327-335.
[15]R.E.Ewing,A coupled nonlinear hyperbolic-Sobolev system,Ann.Mat.Pura Appl.,Ser.Ⅳ,114(1977) 331-349.
[16]R.E.Ewing,Numcrical solution of Sobolcv partial differential equations,SIAM J.Numer.Anal.,12(1975) 345-363.
[17]Lin Yangping,Zhang Tie,Finite element methods for nonlinear Sobolcv equations with nonlinear boundary conditions,Journal of Mathematical Analysis and Applications 165,1992 180-191.
[18]R.E.Ewing,Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation,s,SIAM J.Numcr.Aual.,15(1978) 1125-1150.
[19]朱江,非线性Sobolev方程的有限元法,东北数学,5(2)(1989)179-196.
[20]芮洪兴,拟线性Sobolev方程的特征有限元格式及最优阶误差估计,山东大学学报(自然科学版),29(2)(1994)137-145.
[21]Tongjun Sun,Danping Yang,The finite difference streamline diffusion methods for Sobolev equations with convection dominated term,Applied Mathematics and Computation,125(2002),325-345.
[22]Tongjun Sun,Danping Yang,Error Estimates for a Discontinuous Galerkin Method with Interior Penalties Applied to Nonlinear Sobolev Equations.Numerical Methods for Partial Differential Equation,24(2008),879-896.
[23]孙同军,马克颖,带对流项的拟线性Sobolev方程的差分-流线扩散法,高校计算数学学报,23(4)(2001),340-356.
[24] P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Publ., Amsterdam, 1978.
[25] F.E. Browder, Existence and uniqueness theroems for sloutions of nonlinear boundary value problems, Applications of Partial Differential Equations (R.Finn, ed.), Amercian Mathematical Society, Providence, pp. 24-29, 1965.
[26] M.F.Wheeler.A priori L~2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J.Numer.Anal.,10(4) (1973) 723-758.
[27] P.-A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, In Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, 1977, 292-315, Springer- Verlag,Berlin.
28] J.C. Ncd(?)l(?)c, Mixed finite elements in R~3, Numer. Math. 35(1980), 315-341.
[29] Z. Chen, B. Cockburn, C. Gardner, J.W. Jerome, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys.117(1995), 274-280.
[30] J. Douglas Jr., R.E. Ewing, M. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal.Numer., 17(1983), 17-33.
[31] J. Douglas, Jr. and J.E. Roberts, Global estimates for mixed methods for second order elliptic problems, Math. Comp. 44(1985), 39-52.
[32] E.-J. Park, Mixed finite element methods for nonlinear second order elliptic problems, SIAM J. Numer. Anal. 32(1995), 865-885.
[33] C. Johnson and V. Thom(?)e, Error estimates for some mixed finite element methods for parabolic type problems, R.A.LR.O., Anal. Numer. 14(1981),41-78.
[34] Z. Chen and J. Douglas, Jr., Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems, Mat. Aplic. Comp.10(1991), 137-160.
[35]A.K.Pani,An H~1-Galerkin mixed finite element method for parabolic partial differential equations,SIAM J.Numer.Anal.25(1998),712-727.
[36]A.K.Pani and G.Fairweather,H~1-Galerkin mixed finite element methods for parabolic integro-differential equations,IMA J.Numer.Anal.22(2002),231-252.
[37]A.K.Pani,R.K.Sinha and A.Otta,An H~1-Galerkin mixed finite element methods for second order hyperbolic equations,Inter.J.Numer.Anal.Model.1(2004),111-129.
[38]郭铃,陈焕贞,Sobolev方程的H~1-Galerkin混合有限元方法,系统科学与数学,26(3),2006,301-314.
[39]Pehlivanov A I.Carge G F and Lazarov R D.Least-square mixed finite clement method for second order elliptic problems,SIAM J.Numer.Anal.31(1994):1368-1377.
[40]黄明游,发展方程数值计算方法,科学出版社,2004年6月.
[2]M.Deflour,W.Hager and F.Trochu,Discontinuous Galerkin methods for ordinary differential equations,Math.Comput.36(1981) 455-473.
[3]P.Lcsaint and P.Ravarit,On a finite element method for solving the neutron transport equation,Mathematical Aspects of Finite Elements in Partial Differential Equations,C.de Boor,ed.,Academic Press,New York,1974,89-123.
[4]P.Jamet,Galerkin-type approxiamtions which are discontinuous in time for parabolic equations in a variable domain,SIAM J,Numer.Anal.,15(1978)912-928.
[5]K.Eriksson,C.,Johnson and V.Thomee,Time discretization of parabolic problems by the discontinuous Galerkin method,RAIRO Math.Anal.,19(1985)611-643.
[6]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems I:A linear model problems,SIAM,J,Numer.Anal.,28(1991) 43-77.
[7]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅱ:Optimal error estimates in L_∞L_2 and L_∞L_∞,SIAM J,Numer.Anal.,32(1995) 706-740.
[8]K.Eriksson and C.Johnson,Adaptive finite element methods for parabolic problems Ⅳ:Nonlinear problems,SIAM J,Numer.Anal.,32(1995) 1729-1749.
[9]CH.G.Makridakis and I.Babuska,On the stability of the discontinuous Galerkin method for the heat equation,SIAM J,Numer.Anal.,34(1997) 389-401.
[10]G.I.Barenblatt,I.P.Zheltov and I.N.Kochina,Basic concepts in the theory of seepage of homogenous liquids in fissured rocks,J,Appl.Math.Mech.,24(1960) 1286-1303.
[11]施德明,非线性湿气迁移方程的初边值问题,应用数学学报,13(1990)31-38.
[12]Ting T W,A cooling process according to two-temperature theory of heat conduction.J Math Anal Appl,45(1974) 23-31.
[13]P.J.Chen and M.E.Gurtin,On a theory of heat conduction involving two temperatures,Z.Angew.Math.Phys.,19(1968) 614-627.
[14]P.L.Davis,A quasilinear parabolic and a related third order problem,J.Math.Anal.Appl.,49(1970) 327-335.
[15]R.E.Ewing,A coupled nonlinear hyperbolic-Sobolev system,Ann.Mat.Pura Appl.,Ser.Ⅳ,114(1977) 331-349.
[16]R.E.Ewing,Numcrical solution of Sobolcv partial differential equations,SIAM J.Numer.Anal.,12(1975) 345-363.
[17]Lin Yangping,Zhang Tie,Finite element methods for nonlinear Sobolcv equations with nonlinear boundary conditions,Journal of Mathematical Analysis and Applications 165,1992 180-191.
[18]R.E.Ewing,Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation,s,SIAM J.Numcr.Aual.,15(1978) 1125-1150.
[19]朱江,非线性Sobolev方程的有限元法,东北数学,5(2)(1989)179-196.
[20]芮洪兴,拟线性Sobolev方程的特征有限元格式及最优阶误差估计,山东大学学报(自然科学版),29(2)(1994)137-145.
[21]Tongjun Sun,Danping Yang,The finite difference streamline diffusion methods for Sobolev equations with convection dominated term,Applied Mathematics and Computation,125(2002),325-345.
[22]Tongjun Sun,Danping Yang,Error Estimates for a Discontinuous Galerkin Method with Interior Penalties Applied to Nonlinear Sobolev Equations.Numerical Methods for Partial Differential Equation,24(2008),879-896.
[23]孙同军,马克颖,带对流项的拟线性Sobolev方程的差分-流线扩散法,高校计算数学学报,23(4)(2001),340-356.
[24] P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Publ., Amsterdam, 1978.
[25] F.E. Browder, Existence and uniqueness theroems for sloutions of nonlinear boundary value problems, Applications of Partial Differential Equations (R.Finn, ed.), Amercian Mathematical Society, Providence, pp. 24-29, 1965.
[26] M.F.Wheeler.A priori L~2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J.Numer.Anal.,10(4) (1973) 723-758.
[27] P.-A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, In Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, 1977, 292-315, Springer- Verlag,Berlin.
28] J.C. Ncd(?)l(?)c, Mixed finite elements in R~3, Numer. Math. 35(1980), 315-341.
[29] Z. Chen, B. Cockburn, C. Gardner, J.W. Jerome, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys.117(1995), 274-280.
[30] J. Douglas Jr., R.E. Ewing, M. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal.Numer., 17(1983), 17-33.
[31] J. Douglas, Jr. and J.E. Roberts, Global estimates for mixed methods for second order elliptic problems, Math. Comp. 44(1985), 39-52.
[32] E.-J. Park, Mixed finite element methods for nonlinear second order elliptic problems, SIAM J. Numer. Anal. 32(1995), 865-885.
[33] C. Johnson and V. Thom(?)e, Error estimates for some mixed finite element methods for parabolic type problems, R.A.LR.O., Anal. Numer. 14(1981),41-78.
[34] Z. Chen and J. Douglas, Jr., Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems, Mat. Aplic. Comp.10(1991), 137-160.
[35]A.K.Pani,An H~1-Galerkin mixed finite element method for parabolic partial differential equations,SIAM J.Numer.Anal.25(1998),712-727.
[36]A.K.Pani and G.Fairweather,H~1-Galerkin mixed finite element methods for parabolic integro-differential equations,IMA J.Numer.Anal.22(2002),231-252.
[37]A.K.Pani,R.K.Sinha and A.Otta,An H~1-Galerkin mixed finite element methods for second order hyperbolic equations,Inter.J.Numer.Anal.Model.1(2004),111-129.
[38]郭铃,陈焕贞,Sobolev方程的H~1-Galerkin混合有限元方法,系统科学与数学,26(3),2006,301-314.
[39]Pehlivanov A I.Carge G F and Lazarov R D.Least-square mixed finite clement method for second order elliptic problems,SIAM J.Numer.Anal.31(1994):1368-1377.
[40]黄明游,发展方程数值计算方法,科学出版社,2004年6月.