A Nonconforming Mixed Finite Element Method for Semilinear Pseudo-Hyperbolic Partial Integro-Differential Equations

In this paper, a nonconforming mixed finite element method is studied for semilinear pseudo-hyperbolic partial integrodifferential equations. By use of the interpolation technique instead of the generalized elliptic projection, the optimal error estimates of the corresponding unknown function are given.

• Jingbo Yang
• Hong Li
• Yang Liu
• Siriguleng He

• Pseudo-hyperbolic partial integro-differential equations
• Nonconforming mixed element method
• Semilinear
• Error estimates.

[1] J Nagumo, S Arimoto, S Yoshizawa. An active pulse transmission line
simulating nerve axon, Proc. IRE., 1962, 50: 91-102.
[2] C V Pao. A mixed initial boundary value problem arising in neurophysiology,
J. Math. Anal. Appl., 1975, 52: 105-119.
[3] X Cui. Sobolev-Volterra projection and numerical analysis of finite
element methods for integro-differential equations, Acta Mathematicae
Applicatae Sinica, 2001, 24(3): 441-454.
[4] Y Liu, H Li, W Gao, S He. A new splitting positive definite mixed element
method for pseudo-hyperbolic equations, Mathematica Applicata,
2011, 24(1): 104-111.
[5] 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).
[6] Y Liu, H Li. H1-Galerkin mixed finite element methods for pseudohyperbolic
equations, Appl. Math. Comput., 2009, 212: 446-457.
[7] Y Liu, J F Wang, H Li, W Gao, S He. A new splitting H1-Galerkin
mixed method for pseudo-hyperbolic equations, International Journal of
Engineering and Natural Sciences, 2011, 5(2): 58-63.
[8] Y Liu, H Li, S He. Error estimates of H1-Galerkin mixed finite element
methods for pseudo-hyperbolic partial integro-differential equation [J].
Numerical Mathematics A Journal of Chinese Universities, 2010, 32(1):
1-20.(in Chinese)
[9] Y Liu, H Li. A new mixed finite element method for pseudo-hyperbolic
equation, Mathematica Applicata, 2010, 23(1): 150-157.
[10] H Guo, H X Rui. Least-squares Galerkin procedures for pseudohyperbolic
equations, Appl. Math. Comput., 2007, 189: 425-439.
[11] J Jr Douglas, R Ewing, M Wheeler. A time-discretization procedure
for a mixed finite element approximation of miscible displacement in
porous media, RAIRO Anal. Num'er., 1983, 17: 249-265.
[12] F Brezzi, J Jr Douglas, L Marini. Two families of mixed finite elements
for second order elliptic problems, Numer. Math., 1985, 47: 217-235.
[13] F Brezzi, J Jr Douglas, R Dur'an, M Fortin. Mixed finite elements for
second order elliptic problems in three variables, Numer. Math., 1987,
51: 237-250.
[14] Z D Luo. Theory Bases and Applications of Mixed Finite Element
Methods, Science Press, Beijing, 2006.(in Chinese)
[15] Y P Chen, Y Q Huang. The superconvergence of mixed finite element
methods for nonlinear hyperbolic equations, Communications in Nonlinear
Science and Numerical Simulation, 1998, 3(3): 155-158.
[16] D P Yang. A splitting positive definite mixed element method for
miscible displacement of compressible flow in porous media, Numerical
Methods for Partial Differential Equations, 2001, 17: 229-249.
[17] G Ma, D Y Shi. The nonconforming mixed finite element method for
generalized nerve conduction type equation, Mathematics In Practice
And Theory, 2010, 40(4): 217-223.(in Chinese)
[18] D Y Shi, H B Guan. A kind of full-discrete nonconforming finite element
method for the parabolic variational inequality, Acta Mathematicae
Applicatae Sinica, 2008, 31(1): 90-96.
[19] D Y Shi, J C Ren. Nonconforming mixed finite element method for
the stationary conduction-convection problem, Inter. J. Numer. Anal.
Model., 2009, 6(2): 293-310.
[20] A K Pani, R K Sinha, A K Otta. An H1-Galerkin mixed method for
second order hyperbolic equations, Inter. J. Numer. Anal. Model., 2004,
1(2): 111-129.
[21] Z X Chen. Expanded mixed finite element methods for linear second
order elliptic problems I, RAIRO Mod'el. Math. Anal. Num'er., 1998,
32(4): 479-499.