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Zhou, Zhaojie; Wang, Weiwei; Chen, Huanzhen

An \(H^1\)-Galerkin expanded mixed finite element approximation of second-order nonlinear hyperbolic equations. (English) Zbl 1470.65170

Abstr. Appl. Anal. 2013, Article ID 657952, 12 p. (2013).
Summary: We investigate an \(H^1\)-Galerkin expanded mixed finite element approximation of nonlinear second-order hyperbolic equations, which model a wide variety of phenomena that involve wave motion or convective transport process. This method possesses some features such as approximating the unknown scalar, its gradient, and the flux function simultaneously, the finite element space being free of LBB condition, and avoiding the difficulties arising from calculating the inverse of coefficient tensor. The existence and uniqueness of the numerical solution are discussed. Optimal-order error estimates for this method are proved without introducing curl operator. A numerical example is also given to illustrate the theoretical findings.

Cited in 5 Documents

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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\textit{Z. Zhou} et al., Abstr. Appl. Anal. 2013, Article ID 657952, 12 p. (2013; Zbl 1470.65170)
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References:

[1] Baker, G. A., Error estimates for finite element methods for second order hyperbolic equations, SIAM Journal on Numerical Analysis, 13, 4, 564-576 (1976) · Zbl 0345.65059
[2] Cowsar, L. C.; Dupont, T. F.; Wheeler, M. F., A priori estimates for mixed finite element methods for the wave equation, Computer Methods in Applied Mechanics and Engineering, 82, 1-3, 205-222 (1990) · Zbl 0724.65087
[3] Cowsar, L. C.; Dupont, T. F.; Wheeler, M. F., A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions, SIAM Journal on Numerical Analysis, 33, 2, 492-504 (1996) · Zbl 0859.65097
[4] Geveci, T., On the application of mixed finite element methods to the wave equations, Mathematical Modelling and Numerical Analysis, 22, 2, 243-250 (1988) · Zbl 0646.65083
[5] Chen, H. Z.; Wang, H., An optimal-order error estimate on an \(H^1\)-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow, Numerical Methods for Partial Differential Equations, 26, 1, 188-205 (2010) · Zbl 1425.65094
[6] Pani, A. K.; Fairweather, G., \(H^1\)-Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA Journal of Numerical Analysis, 22, 2, 231-252 (2002) · Zbl 1008.65101
[7] Pani, A. K.; Manickam, S. A. V.; Moudgalya, K. K., Higher order fully discrete scheme combined with \(H^1\)-Galerkin mixed finite element method for semilinear reaction-diffusion equations, Journal of Applied Mathematics and Computing, 15, 1-2, 1-28 (2004) · Zbl 1053.65081
[8] Pani, A. K., An \(H^1\)-Galerkin mixed finite element method for parabolic partial differential equations, SIAM Journal on Numerical Analysis, 35, 2, 712-727 (1998) · Zbl 0915.65107
[9] Guo, L.; Chen, H. Z., \(H^1\)-Galerkin mixed finite element method for the regularized long wave equation, Computing, 77, 2, 205-221 (2006) · Zbl 1098.65096
[10] Pani, A. K.; Sinha, R. K.; Otta, A. K., An \(H^1\)-Galerkin mixed method for second order hyperbolic equations, International Journal of Numerical Analysis and Modeling, 1, 2, 111-129 (2004) · Zbl 1082.65100
[11] Arnold, D. N.; Scott, L. R.; Vogelius, M., Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4, 15, 2, 169-192 (1988) · Zbl 0702.35208
[12] Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Computational Mathematics, 5 (1986), New York, NY, USA: Springer, New York, NY, USA · Zbl 0585.65077
[13] Raviart, P. A.; Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods. Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, 606, 292-315 (1977), New York, NY, USA: Springer, New York, NY, USA · Zbl 0362.65089
[14] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Analyse Numérique, 8, 2, 129-151 (1974) · Zbl 0338.90047
[15] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15 (1991), New York, NY, USA: Springer, New York, NY, USA · Zbl 0788.73002
[16] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, 4 (1978), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0383.65058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.