非线性发展方程的H~1-Galerkin扩展混合有限元方法
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摘要
在工农业生产及其科学研究中,大量的实际问题可由非线性(线性)发展方程刻画,如非线性(线性)声波问题,声学问题,环境流体流动问题等.对该类问题的数值模拟已成为应用数学，计算数学和工程科学研究的热点之一.本文就一类刻画非线性(线性)波动的发展方程-二阶双曲型方程,针对工程应用中对声压(未知函数),声压梯度(未知函数梯度)和波动加速度(伴随向量)的关注,提出了能同时高精度逼近声压,声压梯度和波动加速度的H1-Galerkin扩展混合有限元方法,并从数值分析理论角度证明了该方法是模拟该类问题的高性能数值模拟技术.
首先讨论用于刻画线性波的二阶双曲方程初边值问题提出了数值模拟该类问题的H1-Galerkin扩展混合有限元方法.证明了变分形式与边值问题的等价性和离散格式解的存在唯一性，并导出了离散解对真解的最优收敛精度.理论分析表明，该方法可兼具H1-Galerkin方法和扩展混合有限元方法的优点,即离散格式能同时高精度逼近未知函数，未知函数的梯度和伴随向量函数,有限元空间无需要求满足LBB相容性条件且允许不同变量的逼近空间取不同次数的多项式,同时该方法可以避免计算过程中对系数α(x,t)求逆：保证了在小系数α(x,t)的情形下逼近的有效性.
鉴于实际的波动形式大都是以非线性波形式出现,因此我们进一步讨论了刻画非线性波的二阶非线性双曲方程初边值问题提出了数值模拟该类问题的H1-Galerkin扩展混合有限元方法.证明了变分形式与边值问题的等价性和离散格式解的存在唯一性,并利用归纳假设导出了离散解对真解的最优L2收敛精度.理论分析表明,该方法可同时高精度逼近声压p,声压梯度▽p和波动加速度α(p)▽p,是一种数值模拟非线性波动问题的有效方法.另外,我们注意到该方法可避免对α(p)求逆,从而保证了高密度介质情形下对非线性波问题逼近的有效性,同时克服了H1-Galerkin混合元方法由于对非线性系数α(p)关于时间求导所导致的数值分析困难.
Many practical phenomenons,such as nonlinear(linear) wave problems,acous-tic problems,environmental fluid mechanics,are governed by nonlinear(linear)evo-lution equations,so the numerical simulation for such problems become a hot spot in applied mathematics,computational mathematics and engineering science.Sec-ond order hyperbolic equation is a special kind of evolution equations,which de-scribes such important phenomenons as nonlinear(linear)wave and acoustic prob-lems.Based on the special concerns on the sound pressure (the unknown scalar), the gradient of sound pressure(the gradient of unknown scalar)and the acceleration of sound transmission(the adjoint vector)in engineering application, in this paper we propose an H1-Galerkin expanded mixed finite element method for second order hyperbolic equations and intend to approximate the desired three quantities simul-taneously with high accuracy. The numerical analysis verify that this new method is an ideal numerical method for this equation.
We first discuss the following second order linear hyperbolic initial-boundary value problem which is used to govern linear wave. An H1-Galerkin expanded mixed formulation is proposed,the equivalence of the weak form and the initial-boundary value problem is proved,and the existence and uniqueness of discrete solutions is given.And then the optimal L2 estimates of three variables are derived.This shows this new method have both merits of the expanded mixed finite element method and H1-Galerkin method,that is,the optimal estimates for the unknown scalar,the gradient of unknown scalar and the adjoint vector,being free of LBB stability condition and the finite element spaces having different polynomial degrees.Another advantage of this method we find so far is no need to invert the coefficient tensor, which assures the applicability for problem with small coefficient.
In fact,the most of the sound transmissions are in nonlinear form,so we further discuss the following nonlinear hyperbolic problem An H1-Galerkin expanded mixed formulation is proposed,the equivalence between the weak form and the initial-boundary value problem is proved,and the existence and uniqueness of discrete solutions is given.And then the optimal L2 estimates of three variables are derived by an inductive hypothesis.The theoretical analysis show that the method is a high-performance method for the nonlinear wave equa-tion.Also we find that the new method is no need to invert the coefficient tensor and can simulate the transmission in high density media, and this formulation over-comes the difficulties resulted from the differentiation of a(p)with respect to time t as done in H1-Galerkin mixed finite element method.
首先讨论用于刻画线性波的二阶双曲方程初边值问题提出了数值模拟该类问题的H1-Galerkin扩展混合有限元方法.证明了变分形式与边值问题的等价性和离散格式解的存在唯一性，并导出了离散解对真解的最优收敛精度.理论分析表明，该方法可兼具H1-Galerkin方法和扩展混合有限元方法的优点,即离散格式能同时高精度逼近未知函数，未知函数的梯度和伴随向量函数,有限元空间无需要求满足LBB相容性条件且允许不同变量的逼近空间取不同次数的多项式,同时该方法可以避免计算过程中对系数α(x,t)求逆：保证了在小系数α(x,t)的情形下逼近的有效性.
鉴于实际的波动形式大都是以非线性波形式出现,因此我们进一步讨论了刻画非线性波的二阶非线性双曲方程初边值问题提出了数值模拟该类问题的H1-Galerkin扩展混合有限元方法.证明了变分形式与边值问题的等价性和离散格式解的存在唯一性,并利用归纳假设导出了离散解对真解的最优L2收敛精度.理论分析表明,该方法可同时高精度逼近声压p,声压梯度▽p和波动加速度α(p)▽p,是一种数值模拟非线性波动问题的有效方法.另外,我们注意到该方法可避免对α(p)求逆,从而保证了高密度介质情形下对非线性波问题逼近的有效性,同时克服了H1-Galerkin混合元方法由于对非线性系数α(p)关于时间求导所导致的数值分析困难.
Many practical phenomenons,such as nonlinear(linear) wave problems,acous-tic problems,environmental fluid mechanics,are governed by nonlinear(linear)evo-lution equations,so the numerical simulation for such problems become a hot spot in applied mathematics,computational mathematics and engineering science.Sec-ond order hyperbolic equation is a special kind of evolution equations,which de-scribes such important phenomenons as nonlinear(linear)wave and acoustic prob-lems.Based on the special concerns on the sound pressure (the unknown scalar), the gradient of sound pressure(the gradient of unknown scalar)and the acceleration of sound transmission(the adjoint vector)in engineering application, in this paper we propose an H1-Galerkin expanded mixed finite element method for second order hyperbolic equations and intend to approximate the desired three quantities simul-taneously with high accuracy. The numerical analysis verify that this new method is an ideal numerical method for this equation.
We first discuss the following second order linear hyperbolic initial-boundary value problem which is used to govern linear wave. An H1-Galerkin expanded mixed formulation is proposed,the equivalence of the weak form and the initial-boundary value problem is proved,and the existence and uniqueness of discrete solutions is given.And then the optimal L2 estimates of three variables are derived.This shows this new method have both merits of the expanded mixed finite element method and H1-Galerkin method,that is,the optimal estimates for the unknown scalar,the gradient of unknown scalar and the adjoint vector,being free of LBB stability condition and the finite element spaces having different polynomial degrees.Another advantage of this method we find so far is no need to invert the coefficient tensor, which assures the applicability for problem with small coefficient.
In fact,the most of the sound transmissions are in nonlinear form,so we further discuss the following nonlinear hyperbolic problem An H1-Galerkin expanded mixed formulation is proposed,the equivalence between the weak form and the initial-boundary value problem is proved,and the existence and uniqueness of discrete solutions is given.And then the optimal L2 estimates of three variables are derived by an inductive hypothesis.The theoretical analysis show that the method is a high-performance method for the nonlinear wave equa-tion.Also we find that the new method is no need to invert the coefficient tensor and can simulate the transmission in high density media, and this formulation over-comes the difficulties resulted from the differentiation of a(p)with respect to time t as done in H1-Galerkin mixed finite element method.
引文
[1]Gary A.Sod.A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws[J].Comput. Phys.,1978,27(1):1-31.
[2]V. Thomee. Finite differences for linear parabolic equations[J].Numerical Analysis I,1991:5-196.
[3]Li Ronghua. Generalized Difference Methods for a Nonlinear Dirichlet Problem [J]. SIAM.J.Numer.Anal,1987,24(1):77-88.
[4]E.Greenwell-Yanik, G.Fairweather.Finite element methods for parabolic and hy-perbolic integro-differential equations[J].Nonlinear Anal.Theor. Meth.Appl.,1988, 12(1):785-809.
[5]Zhang Tie,Lin Yanping.L∞-error bounds for some nonlinear integro-differential equations by finite element approximations [J].Journal of Computation Mathemati-cal,1990,12(1):260-269.
[6]V.Thomee.Galerkin Finite Element Methods for parabolic problems[M]. Lecture Notes in Mathematics,Springer,1984.
[7]G.Fairweather.Finite element Galerkin methods for differential equations[M]. Lec-ture Notes in Pure and Applied Mathematics,1978.
[8]F.Brezzi,M.Fortin.Mixed and hybrid finite element methods[M].Springer-Verlag, New York, Inc.,New York,1991.
[9]S.C.Brenner and L.R.Scott.The mathematical theory of finite element methods[M]. Springer-Verlag,New York,1994.
[10]D.N Arnold,L.R Scott,and M.Vogelius.Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon,Ann.Scuola Norm[J].Sup. Piss Cl.Sci-Serie Ⅳ ⅩⅤ 1988:169-192.
[11]T.Geveci. On the applications of mixed finite element methods to the wave equa-tion[J].Math.Model, Numer.Anal.,1988,22(1):243-250.
[12]P. A.Raviart,J.M.Thomas.A mixed finite element method for 2nd order ellip-tic problems[J].Mathematical Aspect of Finite Element Methods,Lecture Notes in Math.,Springer-Verlag, Berlin,New York,1977,606(1):292-315.
[13]J.C.Nedelec. Mixed finite element in R3[J].Numer.Math.,1980,35:315-341.
[14]F.Brezzi,J.Jr.Douglas, R. Duran, M.Fortin.Mixed finite elements for second order elliptic problems in three space variables[J].Numer. Math.,1987,51:237-250.
[15]F.Brezzi,J.Jr.Douglas,M.Fortin,L.D.Marini.Efficient rectangular mixed finite elements in two and three space variables[J].RAIRO Model.Math.Anal. Numer. 1987,4:581-604.
[16]F.Brezzi,J.Jr.Douglas,L.D.Marini.Two families of mixed finite elements for second order elliptic problems[J].Numer. Math.,1985.47:217-235.
[17]F.Brezzi,J.Douglas,Jr.,R. Duran and M. Fortin.Mixed finite element methods for second order elliptic problems in three space variables[J].Numer. Math.,1987, 51:237-250.
[18]A.K.Pani. An H1-Galerkin mixed finite element methods for second order hyper-bolic equations[J].Numer. Anal.2004:111-129.
[19]Ling Guo.Huanzhen Chen. H1-Galerkin mixed finite element method for regularized long wave equation[J]. Computing,2006,77(2):205-221.
[20]A.K.Pani.An.H1-Galerkin mixed finite element method for parabolic partial differential equations[J].SIAM J.Numer.Anal.,1998,35:712-727.
[21]A.K.Pani and G.Fairweather.H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J].IMA J.Numer.Anal.,2002,22: 231-252.
[22]A.K.Pani,S.A.Veda Manickam, K. M. Moudgalya. Higher order fully discrete scheme combined with H1-Galerkin mixed finite element method for semilinear reaction-diffusion equations[J].Applied Mathematics and Computing,2004,15:1-28.
[23]A.K.Pani,P. C.Das.An H1-Galerkin method for quasi-linear parabolic partial differential equations[J].Methods of Functional Analysis in Approximation Theory, C.A.Micchelli,D.V. Pai,and B.V. Limaye,(eds.),ISNM 76,Birkhauser-Verlag, Basel,1986,357-370.
[24]Chen Zhangxin.Expanded mixed finite element methods for quasi-linear second order elliptic problems[J].Ⅱ,M2AN,1998,32:501-520.
[25]Chen Zhangxin. Expanded mixed finite element methods for linear second-order elliptic problems[J].M2AN,1998,32(4):479-499.
[26]Huanzhen Chen, Hong Wang. An optimal-order error estimate on an H1-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow, Numerical Method for partial Differential Equation[J],2010,26(1):188-205.
[27]Ling Guo, Huanzhen Chen. An expanded characteristic mixed finite element method for a convection-dominated transport problem[J].Journal of computational mathe-matics,2005,23(5):479-490.
[28]Huanzhen Chen, Ziwen Jiang.A characteristics-mixed finite element method for Burgers Equation[J].Journal of Applied Mathematics and Computing,2004,15(1-2):29-51.
[29]Randall J.Leveque. Finite Volume Methods for Hyperbolic Problems[M].West Ny-ack:Cambridge University,2002.
[30]R T Beyer.Nonlinear Acoustics[M].Dept.of the Navy, Sea System Company,1974.
[31]F.Brezzi.On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[J].RAIRO Anal.Numer.1974,8:129-151.
[32]J.Douglas,JR. T.F.Dupont,and M.F.Wheeler,H1-Galerkin methods for the Laplace and heat equations,Mathematical Aspects of the Finite Element Methods in Partial Differential Equations,C.de Boor,ed.,Academic Press, New York,1975: 383-415.
[33]P. G.Ciarlet.The Finite Element Method for Elliptic Problems[M].North-Holland, Amsterdam,1978.
[34]袁益让,王宏.非线性双曲型方程有限元方法的误差估计[J].系统科学与数学.1985,5(3)：161-171.
[35]王宏.关于非线性双曲型方程全离散有限元方法的稳定性和收敛性估计[J].计算数学,1987,9(2)：163-175.
[36]罗振东.有限元混合法理论基础及其应用[M].山东教育出社,1996.
[37]郭玲,陈焕贞.Sobolev方程的H1-Galerkin混合有限元方法[J].系统科学与数学,2006,26(3)：301-314.
[38]周兆杰,陈焕贞.伪抛物型积分微分方程的H1-Galerkin混合有限元数值模拟[J].山东师范大学学报,2005,20(2)：3-8.
[39]朱爱玲,杨青.线性抛物型积分-微分方程的扩展混合元方法[J].山东师范大学学报,2005,19(2)：10-14.
[40]刘中艳,陈焕贞.二阶抛物问题的扩展混合元方法[J].山东师范大学学报,2005,20(4)：1-5.
[41]周兆杰,陈焕贞.平面二维水沙模型的特征有限元数值模拟[J].高等学校计算数学学报.2007,29(3)：245-256.
[42]周兆杰,陈焕贞.非定常二维Navier-Stokes方程流函数-涡度形式的特征-混合有限元数值模拟[J].应用数学,2007,29(1)：12-18.
[43]陈凤欣.对流占优问题的扩展混合元数值模拟及其理论分析[D].硕士学位论文,山东师范大学,2007.
[44]李太宝.计算声学：声场的方程和计算方法[M].科学出版社,1987：1-30.
[45]钱祖文.非线性声学[M].北京：科学出版社,1992：24-150.
[46]王元明,管平.线性偏微分方程引论[M].东南大学出版社,2002：154-160.
[47]李大潜,陈韵梅.非线性发展方程[M].科学出版社,1989：49-146.
[2]V. Thomee. Finite differences for linear parabolic equations[J].Numerical Analysis I,1991:5-196.
[3]Li Ronghua. Generalized Difference Methods for a Nonlinear Dirichlet Problem [J]. SIAM.J.Numer.Anal,1987,24(1):77-88.
[4]E.Greenwell-Yanik, G.Fairweather.Finite element methods for parabolic and hy-perbolic integro-differential equations[J].Nonlinear Anal.Theor. Meth.Appl.,1988, 12(1):785-809.
[5]Zhang Tie,Lin Yanping.L∞-error bounds for some nonlinear integro-differential equations by finite element approximations [J].Journal of Computation Mathemati-cal,1990,12(1):260-269.
[6]V.Thomee.Galerkin Finite Element Methods for parabolic problems[M]. Lecture Notes in Mathematics,Springer,1984.
[7]G.Fairweather.Finite element Galerkin methods for differential equations[M]. Lec-ture Notes in Pure and Applied Mathematics,1978.
[8]F.Brezzi,M.Fortin.Mixed and hybrid finite element methods[M].Springer-Verlag, New York, Inc.,New York,1991.
[9]S.C.Brenner and L.R.Scott.The mathematical theory of finite element methods[M]. Springer-Verlag,New York,1994.
[10]D.N Arnold,L.R Scott,and M.Vogelius.Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon,Ann.Scuola Norm[J].Sup. Piss Cl.Sci-Serie Ⅳ ⅩⅤ 1988:169-192.
[11]T.Geveci. On the applications of mixed finite element methods to the wave equa-tion[J].Math.Model, Numer.Anal.,1988,22(1):243-250.
[12]P. A.Raviart,J.M.Thomas.A mixed finite element method for 2nd order ellip-tic problems[J].Mathematical Aspect of Finite Element Methods,Lecture Notes in Math.,Springer-Verlag, Berlin,New York,1977,606(1):292-315.
[13]J.C.Nedelec. Mixed finite element in R3[J].Numer.Math.,1980,35:315-341.
[14]F.Brezzi,J.Jr.Douglas, R. Duran, M.Fortin.Mixed finite elements for second order elliptic problems in three space variables[J].Numer. Math.,1987,51:237-250.
[15]F.Brezzi,J.Jr.Douglas,M.Fortin,L.D.Marini.Efficient rectangular mixed finite elements in two and three space variables[J].RAIRO Model.Math.Anal. Numer. 1987,4:581-604.
[16]F.Brezzi,J.Jr.Douglas,L.D.Marini.Two families of mixed finite elements for second order elliptic problems[J].Numer. Math.,1985.47:217-235.
[17]F.Brezzi,J.Douglas,Jr.,R. Duran and M. Fortin.Mixed finite element methods for second order elliptic problems in three space variables[J].Numer. Math.,1987, 51:237-250.
[18]A.K.Pani. An H1-Galerkin mixed finite element methods for second order hyper-bolic equations[J].Numer. Anal.2004:111-129.
[19]Ling Guo.Huanzhen Chen. H1-Galerkin mixed finite element method for regularized long wave equation[J]. Computing,2006,77(2):205-221.
[20]A.K.Pani.An.H1-Galerkin mixed finite element method for parabolic partial differential equations[J].SIAM J.Numer.Anal.,1998,35:712-727.
[21]A.K.Pani and G.Fairweather.H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations[J].IMA J.Numer.Anal.,2002,22: 231-252.
[22]A.K.Pani,S.A.Veda Manickam, K. M. Moudgalya. Higher order fully discrete scheme combined with H1-Galerkin mixed finite element method for semilinear reaction-diffusion equations[J].Applied Mathematics and Computing,2004,15:1-28.
[23]A.K.Pani,P. C.Das.An H1-Galerkin method for quasi-linear parabolic partial differential equations[J].Methods of Functional Analysis in Approximation Theory, C.A.Micchelli,D.V. Pai,and B.V. Limaye,(eds.),ISNM 76,Birkhauser-Verlag, Basel,1986,357-370.
[24]Chen Zhangxin.Expanded mixed finite element methods for quasi-linear second order elliptic problems[J].Ⅱ,M2AN,1998,32:501-520.
[25]Chen Zhangxin. Expanded mixed finite element methods for linear second-order elliptic problems[J].M2AN,1998,32(4):479-499.
[26]Huanzhen Chen, Hong Wang. An optimal-order error estimate on an H1-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow, Numerical Method for partial Differential Equation[J],2010,26(1):188-205.
[27]Ling Guo, Huanzhen Chen. An expanded characteristic mixed finite element method for a convection-dominated transport problem[J].Journal of computational mathe-matics,2005,23(5):479-490.
[28]Huanzhen Chen, Ziwen Jiang.A characteristics-mixed finite element method for Burgers Equation[J].Journal of Applied Mathematics and Computing,2004,15(1-2):29-51.
[29]Randall J.Leveque. Finite Volume Methods for Hyperbolic Problems[M].West Ny-ack:Cambridge University,2002.
[30]R T Beyer.Nonlinear Acoustics[M].Dept.of the Navy, Sea System Company,1974.
[31]F.Brezzi.On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[J].RAIRO Anal.Numer.1974,8:129-151.
[32]J.Douglas,JR. T.F.Dupont,and M.F.Wheeler,H1-Galerkin methods for the Laplace and heat equations,Mathematical Aspects of the Finite Element Methods in Partial Differential Equations,C.de Boor,ed.,Academic Press, New York,1975: 383-415.
[33]P. G.Ciarlet.The Finite Element Method for Elliptic Problems[M].North-Holland, Amsterdam,1978.
[34]袁益让,王宏.非线性双曲型方程有限元方法的误差估计[J].系统科学与数学.1985,5(3)：161-171.
[35]王宏.关于非线性双曲型方程全离散有限元方法的稳定性和收敛性估计[J].计算数学,1987,9(2)：163-175.
[36]罗振东.有限元混合法理论基础及其应用[M].山东教育出社,1996.
[37]郭玲,陈焕贞.Sobolev方程的H1-Galerkin混合有限元方法[J].系统科学与数学,2006,26(3)：301-314.
[38]周兆杰,陈焕贞.伪抛物型积分微分方程的H1-Galerkin混合有限元数值模拟[J].山东师范大学学报,2005,20(2)：3-8.
[39]朱爱玲,杨青.线性抛物型积分-微分方程的扩展混合元方法[J].山东师范大学学报,2005,19(2)：10-14.
[40]刘中艳,陈焕贞.二阶抛物问题的扩展混合元方法[J].山东师范大学学报,2005,20(4)：1-5.
[41]周兆杰,陈焕贞.平面二维水沙模型的特征有限元数值模拟[J].高等学校计算数学学报.2007,29(3)：245-256.
[42]周兆杰,陈焕贞.非定常二维Navier-Stokes方程流函数-涡度形式的特征-混合有限元数值模拟[J].应用数学,2007,29(1)：12-18.
[43]陈凤欣.对流占优问题的扩展混合元数值模拟及其理论分析[D].硕士学位论文,山东师范大学,2007.
[44]李太宝.计算声学：声场的方程和计算方法[M].科学出版社,1987：1-30.
[45]钱祖文.非线性声学[M].北京：科学出版社,1992：24-150.
[46]王元明,管平.线性偏微分方程引论[M].东南大学出版社,2002：154-160.
[47]李大潜,陈韵梅.非线性发展方程[M].科学出版社,1989：49-146.