Abstract
In this paper, a finite element Galerkin method is applied to multi-dimensional Sobolev equations with Burgers’ type nonlinearity and zero forcing function. Some a priori estimates for the exact solution, which are valid uniformly in time as \(t\rightarrow \infty \) and even uniformly in the coefficient of dispersion \(\mu \) as \(\mu \rightarrow 0,\) are derived. Further, optimal error estimates for semidiscrete Galerkin approximations in \(L^{\infty }(L^2)\) and \(L^{\infty }(H^1)\)-norms are established, which again preserve the exponential decay property. Finally, some numerical experiments are conducted which confirm our theoretical findings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cuesta, C.M., Pop, I.S.: Numerical schemes for a pseudo-parabolic Burgers equation: discontinuous data and long-time behaviour. J. Comp. Appl. Math. 224, 268–283 (2009)
Fan, Y., Pop, I.S.: A class of pseudo-parabolic equations: existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization. Math. Methods Appl. Sci. 34, 2329–2339 (2011)
Mikelic, A.: A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure. J. Differ. Equ. 248, 1561–1577 (2010)
Cao, X., Pop, I.S.: Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media. Appl. Math. Lett. 46, 25–30 (2015)
Showalter, R.E.: A nonlinear Parabolic-Sobolev equation. J. Math. Anal. Appl. 50, 183–190 (1975)
Showalter, R.E.: The Sobolev equation I. Appl. Anal. 5, 15–22 (1975)
Showalter, R.E.: The Sobolev equation II. Appl. Anal. 5, 81–99 (1975)
Showalter, R.E.: Sobolev equations for nonlinear dispersive systems. Appl. Anal. 7, 297–308 (1975)
Arnold, D.N., Douglas Jr., J., Thomeé, V.: Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable. Math. Comp. 36, 53–63 (1981)
Nakao, M.T.: Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension. Numer. Math. 47, 139–157 (1985)
Liu, T., Lin, Y.-P., Rao, M., Cannon, J.R.: Finite element methods for Sobolev equations. J. Comp. Math. 20, 627–642 (2002)
Ewing, R.E.: Numerical solution of Sobolev partial differential equations. SIAM J. Numer. Anal. 12, 345–363 (1975)
Ewing, R.E.: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal. 15, 1125–1150 (1978)
Lin, Y-P.: Galerkin methods for nonlinear Sobolev equations. Aequationes Math. 40, 54–66 (1990)
Lin, Y.-P., Zhang, T.: Finite element methods for nonlinear Sobolev equations with nonlinear boundary condition. J. Math. Anal. Appl. 165, 180–191 (1992)
Bajpai, S., Nataraj, N., Pani, A.K., Damazio, P., Yuan, J.Y.: Semidiscrete Galerkin method for equations of motion arising in Kelvin-Voigt model of viscoelastic fluid flow. Numer. Methods PDE 29, 857–883 (2013)
Foias, C., Manley, O., Rosa, R., Temam, R.: Navier-Stokes equations and turbulence. Cambridge University Press, Cambridge (2001)
Acknowledgements
Both authors thanks Professor Amiya K. Pani for his valuable suggestion.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Pany, A.K., Kundu, S. (2018). Optimal Error Estimates for Semidiscrete Galerkin Approximations to Multi-dimensional Sobolev Equations with Burgers’ Type Nonlinearity. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. NAO 2017. Springer Proceedings in Mathematics & Statistics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-319-90026-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-90026-1_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90025-4
Online ISBN: 978-3-319-90026-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)