A C1-Conforming Finite Element Method for Nonlinear Fourth-Order Hyperbolic Equation
In this paper, the C1-conforming finite element method is analyzed for a class of nonlinear fourth-order hyperbolic partial differential equation. Some a priori bounds are derived using Lyapunov functional, and existence, uniqueness and regularity for the weak solutions are proved. Optimal error estimates are derived for both semidiscrete and fully discrete schemes.
[1] T. ZHANG, Finite element analysis for Cahn-Hilliard equation, Mathematica
Numerica Sinica, 2006, 28(3): 281-292.
[2] R. AN, K.T LI, Stabilized finite element approximation for fourth order
obstacle problem[J]. Acta Mathematicae Applicatae Sinica, 2009, 32(6):
1068-1078.
[3] D.Y. SHI, Y.C. PENG. The finite element methods for fourth order
eigenvalue problems on anisotropic meshes, Chinese Journal of Engineering
Mathematics, 2008, 25(6): 1029-1034.
[4] Z.X. CHEN, Analysis of expanded mixed methods for fourth-order
elliptic problems, Numer. Methods Partial Differential Equations, 1997,
13: 483-503.
[5] H. LI, Y. LIU, Mixed discontinuous space-time finite element method for
the fourth-order parabolic integro-differential equations, Mathematica
Numerica Sinica, 2007, 29(4): 413-420.
[6] S. He, H. Li, The mixed discontinuous space-time finite element method
for the fourth order linear parabolic equation with generalized boundary
condition, Mathematica Numerica Sinica, 2009, 31(2): 167-178.
[7] J.C. LI, Mixed methods for fourth-order elliptic and parabolic problems
using radial basis functions, Adv. Comput. Math., 2005, 23: 21-30.
[8] J.C. LI, Full-order convergence of a mixed finite element method for
fourth-order elliptic equations, J. Math. Anal. Appl., 1999, 230: 329-
349.
[9] J.C. LI, Optimal convergence analysis of mixed finite element methods
for fourth-order elliptic and parabolic problems, Numer. Methods Partial
Differential Equations, 2006, 22: 884-896.
[10] J.C. LI, Optimal error estimates of mixed finite element methods for a
fourth-order nonlinear elliptic problem, J. Math. Anal. Appl., 2007, 334:
183-195.
[11] S.C. CHEN, M.F. LIU, Z.H. QIAO, An anisotropic nonconforming element
for fourth order elliptic singular perturbation problem, International
Journal of Numerical Analysis and Modeling, 2010, 7(4): 766-784.
[12] P. DANUMJAYA, A.K. PANI, Numerical methods for the extended
fisher-kolmogorov (EFK) equation, International Journal of Numerical
Analysis and Modeling, 2006, 3(2): 186-210.
[13] M.F. WHEELER, A priori L2-error estimates for Galerkin approximations
to parabolic differential equation, SIAM J. Numer. Anal., 1973,
10: 723-749.
[14] P.G. CIARLET, The Finite Element Method for Elliptic Problems,
Amsterdam: North-Holland, 1978.
[1] T. ZHANG, Finite element analysis for Cahn-Hilliard equation, Mathematica
Numerica Sinica, 2006, 28(3): 281-292.
[2] R. AN, K.T LI, Stabilized finite element approximation for fourth order
obstacle problem[J]. Acta Mathematicae Applicatae Sinica, 2009, 32(6):
1068-1078.
[3] D.Y. SHI, Y.C. PENG. The finite element methods for fourth order
eigenvalue problems on anisotropic meshes, Chinese Journal of Engineering
Mathematics, 2008, 25(6): 1029-1034.
[4] Z.X. CHEN, Analysis of expanded mixed methods for fourth-order
elliptic problems, Numer. Methods Partial Differential Equations, 1997,
13: 483-503.
[5] H. LI, Y. LIU, Mixed discontinuous space-time finite element method for
the fourth-order parabolic integro-differential equations, Mathematica
Numerica Sinica, 2007, 29(4): 413-420.
[6] S. He, H. Li, The mixed discontinuous space-time finite element method
for the fourth order linear parabolic equation with generalized boundary
condition, Mathematica Numerica Sinica, 2009, 31(2): 167-178.
[7] J.C. LI, Mixed methods for fourth-order elliptic and parabolic problems
using radial basis functions, Adv. Comput. Math., 2005, 23: 21-30.
[8] J.C. LI, Full-order convergence of a mixed finite element method for
fourth-order elliptic equations, J. Math. Anal. Appl., 1999, 230: 329-
349.
[9] J.C. LI, Optimal convergence analysis of mixed finite element methods
for fourth-order elliptic and parabolic problems, Numer. Methods Partial
Differential Equations, 2006, 22: 884-896.
[10] J.C. LI, Optimal error estimates of mixed finite element methods for a
fourth-order nonlinear elliptic problem, J. Math. Anal. Appl., 2007, 334:
183-195.
[11] S.C. CHEN, M.F. LIU, Z.H. QIAO, An anisotropic nonconforming element
for fourth order elliptic singular perturbation problem, International
Journal of Numerical Analysis and Modeling, 2010, 7(4): 766-784.
[12] P. DANUMJAYA, A.K. PANI, Numerical methods for the extended
fisher-kolmogorov (EFK) equation, International Journal of Numerical
Analysis and Modeling, 2006, 3(2): 186-210.
[13] M.F. WHEELER, A priori L2-error estimates for Galerkin approximations
to parabolic differential equation, SIAM J. Numer. Anal., 1973,
10: 723-749.
[14] P.G. CIARLET, The Finite Element Method for Elliptic Problems,
Amsterdam: North-Holland, 1978.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:50058", author = "Yang Liu and Hong Li and Siriguleng He and Wei Gao and Zhichao Fang", title = "A C1-Conforming Finite Element Method for Nonlinear Fourth-Order Hyperbolic Equation", abstract = "In this paper, the C1-conforming finite element method is analyzed for a class of nonlinear fourth-order hyperbolic partial differential equation. Some a priori bounds are derived using Lyapunov functional, and existence, uniqueness and regularity for the weak solutions are proved. Optimal error estimates are derived for both semidiscrete and fully discrete schemes.
", keywords = "Nonlinear fourth-order hyperbolic equation, Lyapunov functional, existence, uniqueness and regularity, conforming finite element method, optimal error estimates.", volume = "5", number = "8", pages = "1112-5", }
