Hostname: page-component-5d84bcc8dc-crm7g Total loading time: 0 Render date: 2026-09-09T09:32:19.105Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stability and Convergence of Fully Discrete Galerkin-Galerkin FEMs for Porous Medium Flows

Published online by Cambridge University Press:  03 June 2015

Buyang Li ,
Jilu Wang  and
Weiwei Sun
Buyang Li*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, P.R. China
Jilu Wang*
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Weiwei Sun*
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
*
Email:buyangli@nju.edu.cn
Email:jiluwang2-c@my.cityu.edu.hk
Corresponding author.Email:maweiw@math.cityu.edu.hk
Get access

Abstract

The paper is concerned with the unconditional stability and error estimates of fully discrete Galerkin-Galerkin FEMs for the equations of incompressible miscible flows in porous media. We prove that the optimal L2 error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Theoretical analysis is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of the corresponding time-discrete PDEs, which was proposed in our previous work [26, 27]. Numerical results for both two and three-dimensional flow models are presented to confirm our theoretical analysis.

Keywords

65N1265N3035K61Unconditional stabilityoptimal error estimateGalerkin FEMsincompressible miscible flows

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1141 - 1158
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Adams, R.A. and Fournier, J.F., Sobolev spaces, 2003 Elsevier Ltd, Netherlands.Google Scholar
[2]Achdou, Y. and Guermond, J.L., Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 37 (2000), pp. 799–826.Google Scholar
[3]Akrivis, G.D., Dougalis, V.A. and Karakashian, O.A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31–53.Google Scholar
[4]Amaziane, B. and El Ossmani, M., Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods, Numer. Methods Partial Differential Eq., 24 (2008), pp. 799–832.Google Scholar
[5]Bear, J. and Bachmat, Y., Introduction to Modeling of Transport Phenomena in Porous Media, Springer-Verlag, New York, 1990.Google Scholar
[6]Bear, J. and Bachmat, Y., A generalized theory of hydrodynamic dispersion in porous media, Symposium of Haifa, 1967, International Association of Scientific Hydrology, Publication No.72, pp. 7–16.Google Scholar
[7]Chen, H., Zhou, Z. and Wang, H., An optimal-order error estimate for an H 1-Galerkin mixed method for a pressure equation in compressible porous medium flow, Int. J. Numer. Anal. Modeling, 9 (2012), pp. 132–148.Google Scholar
[8]Cannon, J.R. and Lin, Y., Nonclassical H 1 projection and Galerkin methods for nonlinear parabolic integro-differential equations, Calcolo, 25 (1988), pp. 187–201.CrossRefGoogle Scholar
[9]Chen, Z. and Ewing, R., Mathematical analysis for reservoir models, SIAM J. Math. Anal., 30 (1999), 431–453.Google Scholar
[10]Chen, Z. and Hoffmann, K. -H., Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity, Adv. Math. Sci. Appl., 5 (1995), pp. 363–389.Google Scholar
[11]Douglas, J. Jr., The numerical simulation of miscible displacement, Computational Methods in nonlinear Mechanics (Oden, J.T. Ed.), North Holland, Amsterdam, 1980.Google Scholar
[12]Douglas, J. Jr., Ewing, R. and Wheeler, M.F., A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numer., 17 (1983), pp. 249–265.Google Scholar
[13]Douglas, J. Jr., Furtada, F., and Pereira, F., On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs, Comput. Geosciences, 1 (1997), pp. 155–190.Google Scholar
[14]Durán, R.G., On the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method, SIAM J. Numer. Anal., 25 (1988), pp. 989–1001.Google Scholar
[15]Elliott, C.M., and Larsson, S., A finite element model for the time-dependent joule heating problem, Math. Comp., 64 (1995), pp. 1433–1453.CrossRefGoogle Scholar
[16]Ervin, V.J. and Miles, W.W., Approximation of time-dependent viscoelastic fluid flow: SUPG approximation, SIAM J. Numer. Anal., 41(2003), pp. 457–486.Google Scholar
[17]Ewing, R.E., Russell, T.F. and Wheeler, M.F., Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comput. Methods Appl. Mech. Engrg., 47 (1984), pp. 73–92.Google Scholar
[18]Ewing, R.E. and Wheeler, M.F., Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal., 17(1980), pp. 351–365.Google Scholar
[19]Fairweather, G., Ma, H. and Sun, W., Orthogonal spline collocation methods for the stream function-vorticity formulation of the Navier-Stokes equations, Numer. Methods Partial Differential Equations, 24(2008), 449–464.Google Scholar
[20]Feng, X., On existence and uniqueness results for a coupled system modeling miscible displacement in porous media, J. Math. Anal. Appl., 194 (1995), 883–910.Google Scholar
[21]He, Y., The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), pp. 2097–2124.Google Scholar
[22]Hou, Y., Li, B. and Sun, W., Error analysis of splitting Galerkin methods for heat and sweat transport in textile materials, SIAM J. Numer. Anal., 51 (2013), pp. 88–111.CrossRefGoogle Scholar
[23]Guillén-González, F. and Redondo-Neble, M.V., New error estimates for a viscosity-splitting scheme in time for the three-dimensional Navier–Stokes equations, IMA J Numer Anal, 31(2011), 556–579.Google Scholar
[24]Kellogg, B. and Liu, B., The analysis of a finite element method for the Navier–Stokes equations with compressibility, Numer. Math., 87 (2000), pp. 153–170.Google Scholar
[25]Li, B., Mathematical Modelling, Analysis and Computation of Some Complex and Nonlinear Flow Problems, PhD Thesis, City University of Hong Kong, July, 2012.Google Scholar
[26]Li, B. and Sun, W., Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations, Int. J. Numer. Anal. & Modeling, 10 (2013), 622–633.Google Scholar
[27]Li, B. and Sun, W., Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media, SIAM J. Numer. Anal., 51(2013), 1959–1977.Google Scholar
[28]Liu, B., The analysis of a finite element method with streamline diffusion for the compressible Navier–Stokes equations, SIAM J. Numer. Anal., 38 (2000), pp. 1–16.Google Scholar
[29]Mu, M. and Huang, Y., An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations, SIAM J. Numer. Anal., 35 (1998), pp. 1740–1761.Google Scholar
[30]Ma, N., Convergence analysis of miscible displacement in porous media by mixed finite element and orthogonal collocation methods, 2010 International Conference on Computational and Information Sciences, DOI 10.1109/ICCIS.2010.331Google Scholar
[31]Ma, N., Lu, T. and Yang, D., Analysis of incompressible miscible displacement in porous media by characteristics collocation method, Numer. Methods Partial Differential Eq., DOI 10.1002/num.20123Google Scholar
[32]Malta, S.M.C. and Loula, A.F.D., Numerical analysis of finite element methods for miscible displacements in porous media, Numer. Methods Partial Differential Equations, 14 (1998), 519–548.Google Scholar
[33]Ma, H. and Sun, W., Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation, SIAM J. Numer. Anal., 39 (2001), pp. 1380–1394.Google Scholar
[34]Pani, A.K., Yuan, J. and Damazio, P.O., On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one, SIAM J. Numer. Anal., 44 (2006), 804–825Google Scholar
[35]Peaceman, D.W., Fundamentals of Numerical Reservior Simulations, Elsevier, Amsterdam, 1977.Google Scholar
[36]Radu, F., Pop, I.S. and Knabner, P., Order of convergence estimates for an Euler implicit mixed finite element discretization of Richard equation, SIAM J. Numer. Anal., 42 (2004), 1452–1478.Google Scholar
[37]Rannacher, R. and Scott, R., Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 38 (1982), pp. 437–445.Google Scholar
[38]Raviart, P.A. and Thomas, J.M., A mixed finite element method for 2-nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606, Springer-Verlag, 1977, pp. 292–315.Google Scholar
[39]Sanz-Serna, J. M., Methods for the numerical solution of nonlinear Schrödinger equation, Math. Comp., 43 (1984), pp. 21–27.Google Scholar
[40]Sun, S. and Wheeler, M.F., Analysis of discontinuous Galerkin methods for multicomponent reactive transport problems, Comput. Math. Appl., 52(2006), pp. 637–650.Google Scholar
[41]Sun, S. and Wheeler, M.F., Discontinuous Galerkin methods for coupled flow and reactive transport problems, Applied Numer. Math., 52(2005), pp. 273–298.Google Scholar
[42]Sun, W. and Sun, Z., Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials, Numer Math., 120 (2012), pp. 153–187.Google Scholar
[43]Sun, T. and Yuan, Y., An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method, J. Comput. Appl. Math., 228 (2009), pp. 391–411.Google Scholar
[44]Thomée, V., Galerkin finite element methods for parabolic problems, Springer-Verkag Berkub Geudekberg 1997.Google Scholar
[45]Tourigny, Y., Optimal H 1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation, IMA J. Numer. Anal., 11 (1991), pp. 509–523.Google Scholar
[46]Wang, H., An optimal-order error estimate for a family of ELLAM-MFEM approximations to porous medium flow, SIAM J. Numer. Anal., 46 (2008), pp. 2133–2152.Google Scholar
[47]Wang, K., An optimal-order estimate for MMOC-MFEM approximations to porous medium flow, Numer. Methods Partial Differential Eq., 25 (2009), pp. 1283–1302.Google Scholar
[48]Wang, K., He, Y. and Shang, Y., Fully discrete finite element method for the viscoelastic fluid motion equations, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), pp. 665–684.Google Scholar
[49]Wang, K., He, Y. and Feng, X., On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization, Applied Mathematical Modelling, 34 (2010), 4089–4105.Google Scholar
[50]Zhang, Z. and Ma, H., A rational spectral method for the KdV equation on the half line, J. Comput. Appl. Math., 230 (2009), 614–625.Google Scholar
[51]Zhao, W., Convergence analysis of finite element method for the nonstationary thermistor problem, Shandong Daxue Xuebao, 29 (1994), pp. 361–367.Google Scholar
[52]Zlámal, M., Curved elements in the finite element method. I*, SIAM J. Numer. Anal., 10 (1973), 229–240.Google Scholar