{"title":"A Nonconforming Mixed Finite Element Method for Semilinear Pseudo-Hyperbolic Partial Integro-Differential Equations","authors":"Jingbo Yang, Hong Li, Yang Liu, Siriguleng He","volume":68,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":886,"pagesEnd":894,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10420","abstract":"

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.<\/p>\r\n","references":" J Nagumo, S Arimoto, S Yoshizawa. An active pulse transmission line\r\nsimulating nerve axon, Proc. IRE., 1962, 50: 91-102.\r\n C V Pao. A mixed initial boundary value problem arising in neurophysiology,\r\nJ. Math. Anal. Appl., 1975, 52: 105-119.\r\n X Cui. Sobolev-Volterra projection and numerical analysis of finite\r\nelement methods for integro-differential equations, Acta Mathematicae\r\nApplicatae Sinica, 2001, 24(3): 441-454.\r\n Y Liu, H Li, W Gao, S He. A new splitting positive definite mixed element\r\nmethod for pseudo-hyperbolic equations, Mathematica Applicata,\r\n2011, 24(1): 104-111.\r\n Y Liu, H Li, J F Wang, S He. Splitting positive definite mixed\r\nelement methods for pseudo-hyperbolic equations, Numer. Methods\r\nPartial Differential Equations, DOI 10.1002\/num.20650(2010).\r\n Y Liu, H Li. H1-Galerkin mixed finite element methods for pseudohyperbolic\r\nequations, Appl. Math. Comput., 2009, 212: 446-457.\r\n Y Liu, J F Wang, H Li, W Gao, S He. A new splitting H1-Galerkin\r\nmixed method for pseudo-hyperbolic equations, International Journal of\r\nEngineering and Natural Sciences, 2011, 5(2): 58-63.\r\n Y Liu, H Li, S He. Error estimates of H1-Galerkin mixed finite element\r\nmethods for pseudo-hyperbolic partial integro-differential equation [J].\r\nNumerical Mathematics A Journal of Chinese Universities, 2010, 32(1):\r\n1-20.(in Chinese)\r\n Y Liu, H Li. A new mixed finite element method for pseudo-hyperbolic\r\nequation, Mathematica Applicata, 2010, 23(1): 150-157.\r\n H Guo, H X Rui. Least-squares Galerkin procedures for pseudohyperbolic\r\nequations, Appl. Math. Comput., 2007, 189: 425-439.\r\n J Jr Douglas, R Ewing, M Wheeler. A time-discretization procedure\r\nfor a mixed finite element approximation of miscible displacement in\r\nporous media, RAIRO Anal. Num'er., 1983, 17: 249-265.\r\n F Brezzi, J Jr Douglas, L Marini. Two families of mixed finite elements\r\nfor second order elliptic problems, Numer. Math., 1985, 47: 217-235.\r\n F Brezzi, J Jr Douglas, R Dur'an, M Fortin. Mixed finite elements for\r\nsecond order elliptic problems in three variables, Numer. Math., 1987,\r\n51: 237-250.\r\n Z D Luo. Theory Bases and Applications of Mixed Finite Element\r\nMethods, Science Press, Beijing, 2006.(in Chinese)\r\n Y P Chen, Y Q Huang. The superconvergence of mixed finite element\r\nmethods for nonlinear hyperbolic equations, Communications in Nonlinear\r\nScience and Numerical Simulation, 1998, 3(3): 155-158.\r\n D P Yang. A splitting positive definite mixed element method for\r\nmiscible displacement of compressible flow in porous media, Numerical\r\nMethods for Partial Differential Equations, 2001, 17: 229-249.\r\n G Ma, D Y Shi. The nonconforming mixed finite element method for\r\ngeneralized nerve conduction type equation, Mathematics In Practice\r\nAnd Theory, 2010, 40(4): 217-223.(in Chinese)\r\n D Y Shi, H B Guan. A kind of full-discrete nonconforming finite element\r\nmethod for the parabolic variational inequality, Acta Mathematicae\r\nApplicatae Sinica, 2008, 31(1): 90-96.\r\n D Y Shi, J C Ren. Nonconforming mixed finite element method for\r\nthe stationary conduction-convection problem, Inter. J. Numer. Anal.\r\nModel., 2009, 6(2): 293-310.\r\n A K Pani, R K Sinha, A K Otta. An H1-Galerkin mixed method for\r\nsecond order hyperbolic equations, Inter. J. Numer. Anal. Model., 2004,\r\n1(2): 111-129.\r\n Z X Chen. Expanded mixed finite element methods for linear second\r\norder elliptic problems I, RAIRO Mod'el. Math. Anal. Num'er., 1998,\r\n32(4): 479-499.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 68, 2012"}