A New Splitting H1-Galerkin Mixed Method for Pseudo-hyperbolic Equations

Authors: Yang Liu, Jinfeng Wang, Hong Li, Wei Gao, Siriguleng He

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.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1074391

References:

[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.