A New Splitting H1-Galerkin Mixed Method for Pseudo-hyperbolic Equations
A new numerical scheme based on the H1-Galerkin mixed finite element method for a class of second-order pseudohyperbolic equations is constructed. The proposed procedures can be split into three independent differential sub-schemes and does not need to solve a coupled system of equations. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. And the proposed method dose not requires the LBB consistency condition. Finally, some numerical results are provided to illustrate the efficacy of our method.
[1] J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line
simulating nerve axon, Proc. IRE., 50 (1962) 91-102.
[2] C.V. Pao, A mixed initial boundary value problem arising in neurophysiology,
J. Math. Anal. Appl., 52 (1975) 105-119.
[3] R. Arima, Y. Hasegawa, On global solutions for mixed problems of a
semi-linear differential equation,Proc. Jpn. Acad., 39 (1963) 721-725.
[4] G. Ponce, Global existence of small of solutions to a class of nonlinear
evolution equations, Nonlinear Anal., 9 (1985) 399-418.
[5] W.M. Wan, Y.C. Liu, Long time behaviors of solutions for initial
boundary value problem of pseudohyperbolic equations, Acta Math.
Appl. Sin., 22(2) (1999) 311-355.
[6] H. Guo, H.X. Rui, Least-squares Galerkin procedures for pseudohyperbolic
equations, Appl. Math. Comput., 189 (2007) 425-439.
[7] Y. Liu, H. Li, J.F. Wang, S. He, Splitting positive definite mixed
element methods for pseudo-hyperbolic equations, Numer. Methods
Partial Differential Equations, DOI 10.1002/num.20650, 2010.
[8] J.C. Li, Full-order convergence of a mixed finite element method for
fourth-order elliptic equations, J. Math. Anal. Appl., 230 (1999) 329-
349.
[9] Z.X. Chen, Expanded mixed finite element methods for linear second
order elliptic problems I, RAIRO Model. Math. Anal. Num'er., 32 (1998)
479-499.
[10] J. Douglas, R. Ewing, M. Wheeler, A time-discretization procedure for a
mixed finite element approximation of miscible displacement in porous
media, RAIRO Model. Math. Anal. Numer., 17 (1983) 249-265.
[11] C. Johson, V. Thom'ee, Error estimates for some mixed finite element
methods for parabolic type problems, RAIRO Model. Math. Anal.
Numer., 15 (1981) 41-78.
[12] Z.W. Jiang, H.Z. Chen, Error estimates for mixed finite element methods
for sobolev equation, Northeast Math. J., 17 (2001) 301-314.
[13] Z.D. Luo, R.X. Liu, Mixed finite element analysis and numerical solitary
solution for the RLW equation, SIAM J. Numer. Anal., 36 (1998) 89-
104.
[14] J.S. Zhang, D.P. Yang, A splitting positive definite mixed element
method for second-order hyperbolic equations, Numer. Methods Partial
Differential Equations, 25 (2009) 622-636.
[15] L.C. Cowsar, T.F. Dupont, M.F. Wheeler, A priori estimates for mixed
finite element approximations of second-order hyperbolic equations with
absorbing boundary conditions, SIAM J. Numer. Anal., 33 (1996) 492-
504.
[16] D.Y. Shi, W. Gong, The nonconforming finite element approximations
to hyperbolic equation on anisotropic meshes, Mathematica Applicata,
20 (2007) l96-202
[17] Y.P. Chen, Y.Q. Huang, The superconvergence of mixed finite element
methods for nonlinear hyperbolic equations, Communications in Nonlinear
Science and Numerical Simulation, 3 (1998) 155-158.
[18] A.K. Pani, An H1-Galerkin mixed finite element methods for parabolic
partial differential equations, SIAM J. Numer. Anal., 35 (1998) 712-727.
[19] A.K. Pani, G. Fairweather, H1-Galerkin mixed finite element methods
for parabolic partial integro-differential equations, IMA Journal of
Numerical Analysis, 22 (2002) 231-252.
[20] A.K. Pani, R.K. Sinha, A.K. Otta, An H1-Galerkin mixed method for
second order hyperbolic equations, Int. J. Numer. Anal. Model., 1 (2004)
111-129.
[21] Y. Liu, H. Li, J.F. Wang, Error estimates of H1-Galerkin mixed finite
element method for Schr¨odinger equation, Appl. Math. J. Chinese Univ.,
24 (2009) 83-89.
[22] L. Guo, H.Z. Chen, H1-Galerkin mixed finite element method for the
regularized long wave equation, Computing, 77 (2006) 205-221.
[23] M.F. Wheeler, A priori L2−error estimates for Galerkin approximations
to parabolic differential equation, SIAM J. Numer. Anal., 10 (1973) 723-
749.
[1] J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line
simulating nerve axon, Proc. IRE., 50 (1962) 91-102.
[2] C.V. Pao, A mixed initial boundary value problem arising in neurophysiology,
J. Math. Anal. Appl., 52 (1975) 105-119.
[3] R. Arima, Y. Hasegawa, On global solutions for mixed problems of a
semi-linear differential equation,Proc. Jpn. Acad., 39 (1963) 721-725.
[4] G. Ponce, Global existence of small of solutions to a class of nonlinear
evolution equations, Nonlinear Anal., 9 (1985) 399-418.
[5] W.M. Wan, Y.C. Liu, Long time behaviors of solutions for initial
boundary value problem of pseudohyperbolic equations, Acta Math.
Appl. Sin., 22(2) (1999) 311-355.
[6] H. Guo, H.X. Rui, Least-squares Galerkin procedures for pseudohyperbolic
equations, Appl. Math. Comput., 189 (2007) 425-439.
[7] Y. Liu, H. Li, J.F. Wang, S. He, Splitting positive definite mixed
element methods for pseudo-hyperbolic equations, Numer. Methods
Partial Differential Equations, DOI 10.1002/num.20650, 2010.
[8] J.C. Li, Full-order convergence of a mixed finite element method for
fourth-order elliptic equations, J. Math. Anal. Appl., 230 (1999) 329-
349.
[9] Z.X. Chen, Expanded mixed finite element methods for linear second
order elliptic problems I, RAIRO Model. Math. Anal. Num'er., 32 (1998)
479-499.
[10] J. Douglas, R. Ewing, M. Wheeler, A time-discretization procedure for a
mixed finite element approximation of miscible displacement in porous
media, RAIRO Model. Math. Anal. Numer., 17 (1983) 249-265.
[11] C. Johson, V. Thom'ee, Error estimates for some mixed finite element
methods for parabolic type problems, RAIRO Model. Math. Anal.
Numer., 15 (1981) 41-78.
[12] Z.W. Jiang, H.Z. Chen, Error estimates for mixed finite element methods
for sobolev equation, Northeast Math. J., 17 (2001) 301-314.
[13] Z.D. Luo, R.X. Liu, Mixed finite element analysis and numerical solitary
solution for the RLW equation, SIAM J. Numer. Anal., 36 (1998) 89-
104.
[14] J.S. Zhang, D.P. Yang, A splitting positive definite mixed element
method for second-order hyperbolic equations, Numer. Methods Partial
Differential Equations, 25 (2009) 622-636.
[15] L.C. Cowsar, T.F. Dupont, M.F. Wheeler, A priori estimates for mixed
finite element approximations of second-order hyperbolic equations with
absorbing boundary conditions, SIAM J. Numer. Anal., 33 (1996) 492-
504.
[16] D.Y. Shi, W. Gong, The nonconforming finite element approximations
to hyperbolic equation on anisotropic meshes, Mathematica Applicata,
20 (2007) l96-202
[17] Y.P. Chen, Y.Q. Huang, The superconvergence of mixed finite element
methods for nonlinear hyperbolic equations, Communications in Nonlinear
Science and Numerical Simulation, 3 (1998) 155-158.
[18] A.K. Pani, An H1-Galerkin mixed finite element methods for parabolic
partial differential equations, SIAM J. Numer. Anal., 35 (1998) 712-727.
[19] A.K. Pani, G. Fairweather, H1-Galerkin mixed finite element methods
for parabolic partial integro-differential equations, IMA Journal of
Numerical Analysis, 22 (2002) 231-252.
[20] A.K. Pani, R.K. Sinha, A.K. Otta, An H1-Galerkin mixed method for
second order hyperbolic equations, Int. J. Numer. Anal. Model., 1 (2004)
111-129.
[21] Y. Liu, H. Li, J.F. Wang, Error estimates of H1-Galerkin mixed finite
element method for Schr¨odinger equation, Appl. Math. J. Chinese Univ.,
24 (2009) 83-89.
[22] L. Guo, H.Z. Chen, H1-Galerkin mixed finite element method for the
regularized long wave equation, Computing, 77 (2006) 205-221.
[23] M.F. Wheeler, A priori L2−error estimates for Galerkin approximations
to parabolic differential equation, SIAM J. Numer. Anal., 10 (1973) 723-
749.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59049", author = "Yang Liu and Jinfeng Wang and Hong Li and Wei Gao and Siriguleng He", title = "A New Splitting H1-Galerkin Mixed Method for Pseudo-hyperbolic Equations", abstract = "A new numerical scheme based on the H1-Galerkin mixed finite element method for a class of second-order pseudohyperbolic equations is constructed. The proposed procedures can be split into three independent differential sub-schemes and does not need to solve a coupled system of equations. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. And the proposed method dose not requires the LBB consistency condition. Finally, some numerical results are provided to illustrate the efficacy of our method.
", keywords = "Pseudo-hyperbolic equations, splitting system, H1-Galerkin mixed method, error estimates.", volume = "5", number = "3", pages = "427-6", }
