# American Institute of Mathematical Sciences

March  2011, 15(2): 325-341. doi: 10.3934/dcdsb.2011.15.325

## An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems

 1 School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China, China 2 Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, United States 3 Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received  February 2010 Revised  April 2010 Published  December 2010

In this paper we prove an optimal-order error estimate for a family of characteristic mixed method with arbitrary degree of mixed finite element approximations for the numerical solution of transient convection diffusion equations. This paper generalizes the results in [1, 61]. The proof of the main results is carried out via three lemmas, which are utilized to overcome the difficulties arising from the combination of MMOC and mixed finite element methods. Numerical experiments are presented to justify the theoretical analysis.
Citation: Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325
##### References:
 [1] T. Arbogast and M. F. Wheeler, A characteristics-mixed finite element method for advection-dominated transport problems,, SIAM J. Numer. Anal., 32 (1995), 404.  doi: 10.1137/0732017.  Google Scholar [2] D. N. Arnolds, L. R. Scott and M. Vogelus, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygonal,, Ann. Scuola. Norm. Sup. Pisa, (1988), 169.   Google Scholar [3] M. Bause and P. Knabner, Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems,, SIAM J. Numer. Anal., 39 (2002), 1954.  doi: 10.1137/S0036142900367478.  Google Scholar [4] J. P. Benque and J. Ronat, Quelques difficulties des modeles numeriques en hydraulique,, Comp. Meth. Appl. Mech. Engrg., (1982), 471.   Google Scholar [5] P. J. Binning and M. A. Celia, A finite volume Eulerian-Lagrangian localized adjoint method for solution of the contaminant transport equations in two-dimensional multi-phase flow systems,, Water Resour. Res., 32 (1996), 103.  doi: 10.1029/95WR02763.  Google Scholar [6] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,, RAIRO Anal. Numér., 8 (1974), 129.   Google Scholar [7] F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,", Springer Series in Computational Mathematics, 15 (1991).   Google Scholar [8] M. A. Celia, T. F. Russell, I. Herrera and R. E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation,, Advances in Water Resources, 13 (1990), 187.  doi: 10.1016/0309-1708(90)90041-2.  Google Scholar [9] Z. Chen, Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems,, Comput. Methods Appl. Mech. Engrg., 191 (2002), 2509.  doi: 10.1016/S0045-7825(01)00411-X.  Google Scholar [10] Z. Chen, S.-H. Chou and D. Y. Kwak, Characteristic-mixed covolume methods for advection-dominated diffusion problems,, Numerical Linear Algebra with Applications, 13 (2006), 677.  doi: 10.1002/nla.492.  Google Scholar [11] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, 4 (1978).  doi: 10.1016/S0168-2024(08)70178-4.  Google Scholar [12] H. K. Dahle, R. E. Ewing and T. F. Russell, Eulerian-Lagrangian localized adjoint methods for a nonlinear convection-diffusion equation,, Comp. Meth. Appl. Mech. Engrg., 122 (1995), 223.  doi: 10.1016/0045-7825(94)00733-4.  Google Scholar [13] C. N. Dawson, T. F. Russell and M. F. Wheeler, Some improved error estimates for the modified method of characteristics,, SIAM J. Numer. Anal., 26 (1989), 1487.  doi: 10.1137/0726087.  Google Scholar [14] J. Douglas Jr., F. Furtado and F. Pereira, On the numerical simulation of water flooding of hetergeneous petroleum reserviors,, Comput. Geosci., 1 (1997), 155.  doi: 10.1023/A:1011565228179.  Google Scholar [15] J. Douglas, Jr., C.-S. Huang and F. Pereira, The modified method of characteristics with adjusted advection,, Numer. Math., 83 (1999), 353.  doi: 10.1007/s002110050453.  Google Scholar [16] J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures,, SIAM J. Numer. Anal., 19 (1982), 871.  doi: 10.1137/0719063.  Google Scholar [17] M. S. Espedal and R. E. Ewing, Characteristic Petrov-Galerkin subdomain methods for two-phase immiscible flow,, Proceedings of the first world congress on computational mechanics (Austin, 64 (1987), 113.   Google Scholar [18] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).   Google Scholar [19] R. E. Ewing (Ed.), "The Mathematics of Reservoir Simulation,", Research Frontiers in Applied Mathematics 1, (1984).   Google Scholar [20] R. E. Ewing, T. F. Russell and M. F. Wheeler, 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), 73.  doi: 10.1016/0045-7825(84)90048-3.  Google Scholar [21] A. O. Garder, D. W. Peaceman and A. L. Pozzi, Numerical calculations of multidimensional miscible displacement by the method of characteristics,, Soc. Pet. Eng. J., 4 (1964), 26.   Google Scholar [22] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983).   Google Scholar [23] R. W. Healy and T. F. Russell, A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation,, Water Resour. Res., 29 (1993), 2399.  doi: 10.1029/93WR00403.  Google Scholar [24] R. W. Healy and T. F. Russell, Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method,, Adv. Water Res., 21 (1998), 11.   Google Scholar [25] J. M. Hervouet, Applications of the method of characteristics in their weak formulation to solving two-dimensional advection-equations on mesh grids,, in, (1986), 149.   Google Scholar [26] C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems,, RAIRO Anal. Numer., 15 (1981), 41.   Google Scholar [27] X. Li, W. Wu and O. C. Zienkiewicz, Implicit characteristic Galerkin method for convection-diffusion equations,, Int. J. Numer. Meth. Engrg., 47 (2000), 1689.  doi: 10.1002/(SICI)1097-0207(20000410)47:10<1689::AID-NME850>3.0.CO;2-W.  Google Scholar [28] K. W. Morton, A. Priestley and E. Süli, Stability of the Lagrangian-Galerkin method with nonexact integration,, RAIRO Model. Math. Anal. Num., 22 (1988), 625.   Google Scholar [29] J. C. Nédélec, A new family of mixed finite elements in $\mathbf R^3$,, Numerische Mathematik, 50 (1986), 57.  doi: 10.1007/BF01389668.  Google Scholar [30] S. P. Neuman, An Eulerian-Lagrangian numerical scheme for the dispersion-convection equation using conjugate space-time grids,, J. Comp. Phys., 41 (1981), 270.  doi: 10.1016/0021-9991(81)90097-8.  Google Scholar [31] D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation,", Elsevier, (1977).   Google Scholar [32] G. F. Pinder and H. H. Cooper, A numerical technique for calculating the transient position of the saltwater front,, Water Resou. Res., (1970), 875.   Google Scholar [33] O. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations,, Numer. Math., 38 (): 309.  doi: 10.1007/BF01396435.  Google Scholar [34] P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems,, Mathematical Aspects of the Finite Element Method, 606 (1975), 292.   Google Scholar [35] H.-G. Roos, M. Stynes and L. Tobiska, "Numerical Methods for Singularly Perturbed Differential Equations,", Convection-Diffusion and Flow Problems, (1996).   Google Scholar [36] E. Varoglu and W. D. L. Finn, Finite elements incorporating characteristics for one-dimensional diffusion-convection equation,, J. Comput. Phys., 34 (1980), 371.  doi: 10.1016/0021-9991(80)90095-9.  Google Scholar [37] H. Wang, A family of ELLAM schemes for advection-diffusion-reaction equations and their convergence analyses,, Numerical Methods for PDEs, 14 (1998), 739.   Google Scholar [38] H. Wang, An optimal-order error estimate for an ELLAM scheme for two-dimensional linear advection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1338.  doi: 10.1137/S0036142998335686.  Google Scholar [39] H. Wang, An optimal-order error estimate for MMOC and MMOCAA schemes for multidimensional advection-reaction equations,, Numerical Methods for PDEs, 18 (2002), 69.   Google Scholar [40] H. Wang, An optimal-order error estimate for a family of ELLAM-MFEM approximations to porous medium flow,, SIAM J. Numer. Anal., 46 (2008), 2133.  doi: 10.1137/S0036142903428281.  Google Scholar [41] H. Wang and M. Al-Lawatia, A locally conservative Eulerian-Lagrangian control-volume method for transient advection-diffusion equations,, Numerical Methods for Partial Differential Equations, 22 (2005), 577.  doi: 10.1002/num.20106.  Google Scholar [42] H. Wang, H. K. Dahle, R. E. Ewing, M. S. Espedal, R. C. Sharpley and S. Man, An ELLAM scheme for advection-diffusion equations in two dimensions,, SIAM J. Sci. Comput., 20 (1999), 2160.  doi: 10.1137/S1064827596309396.  Google Scholar [43] H. Wang, R. E. Ewing, G. Qin and S. L. Lyons, "An Eulerian-Lagrangian Formulation for Compositional Flow in Porous Media,", The 2006 Society of Petroleum Engineering Annual Technical Conference in San Antonio, (2006), 24.   Google Scholar [44] H. Wang, R. E. Ewing, G. Qin, S. L. Lyons, M. Al-Lawatia and S. Man, A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations,, J. Comput. Phys., 152 (1999), 120.  doi: 10.1006/jcph.1999.6239.  Google Scholar [45] H. Wang, R. E. Ewing and T. F. Russell, Eulerian-Lagrangian localized methods for convection-diffusion equations and their convergence analysis,, IMA J. Numer. Anal., 15 (1995), 405.  doi: 10.1093/imanum/15.3.405.  Google Scholar [46] H. Wang, X. Shi and R. E. Ewing, An ELLAM scheme for multidimensional advection-reaction equations and its optimal-order error estimate,, SIAM. J. Numer. Anal., 38 (2001), 1846.  doi: 10.1137/S0036142999362389.  Google Scholar [47] H. Wang and K. Wang, Uniform estimates for Eulerian-Lagrangian methods for singularly perturbed time-dependent problems,, SIAM J. Numer. Anal., 45 (2007), 1305.  doi: 10.1137/060652816.  Google Scholar [48] K. Wang, A uniformly optimal-order error estimate of an ELLAM scheme for unstady-state advection-diffusion equations,, International Journal of Numerical Analysis and Modeling, 5 (2008), 286.   Google Scholar [49] K. Wang, An optimal-order estimate for MMOC-MFEM approximations to porous medium flow,, Numer. Methods for Partial Differential Equations, 25 (2008), 1283.  doi: 10.1002/num.20397.  Google Scholar [50] K. Wang, A uniform optimal-order estimate for an Eulerian-Lagrangian discontinuous Galerkin method for transient advection-diffusion equations,, Numer. Methods for Partial Differential Equations, 25 (2009), 87.  doi: 10.1002/num.20338.  Google Scholar [51] K. Wang and H. Wang, A uniform estimate for the ELLAM scheme for transport equations,, Numer. Methods for PDEs, 24 (2008), 535.   Google Scholar [52] K. Wang and H. Wang, An optimal-order error estimate to the modified method of characteristics for a degenerate convection-diffusion equation,, International Journal of Numerical Analysis and Modeling, 6 (2009), 217.   Google Scholar [53] K. Wang and H. Wang, A uniform estimate for the MMOC for two-dimensional advection-diffusion equations,, Numer. Methods for PDEs, 26 (2010), 1054.   Google Scholar [54] K. Wang, H. Wang and M. Al-Lawatia, An Eulerian-Lagrangian discontinuous Galerkin method for transient advection-diffusion equations,, Numer. Methods for Partial Differential Equations, 23 (2007), 1343.  doi: 10.1002/num.20223.  Google Scholar [55] K. Wang, H. Wang and M. Al-Lawatia, A CFL-free explicit characteristic interior penalty scheme for linear advection-reaction equations,, Numer. Methods for PDEs, 26 (2010), 561.   Google Scholar [56] K. Wang, H. Wang, M. Al-Lawatia and H. Rui, A family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations and their optimal-order $L^2$ error estimates,, Commun. Comput. Phys., 6 (2009), 203.  doi: 10.4208/cicp.2009.v6.p203.  Google Scholar [57] M. F. Wheeler and C. N. Dawson, An operator-splitting method for advection-diffusion-reaction problems,, MAFELAP Proceedings, (1988), 463.   Google Scholar [58] L. Wu and H. Wang, An Eulerian-Lagrangian single-node collocation method for transient advection-diffusion equations in multiple space dimensions,, Numerical Methods for Partial Differential Equations, 20 (2004), 284.  doi: 10.1002/num.10094.  Google Scholar [59] L. Wu, H. Wang and G. F. Pinder, A nonconventional Eulerian-Lagrangian single-node collocation method with Hermite polynomials for unsteady-state advection-diffusion equations,, Numerical Methods for PDEs, 19 (2003), 271.   Google Scholar [60] L. Wu and K. Wang, A single-node characteristic collocation method for unsteady-state convection-diffusion equations in three-dimensional spaces,, Numerical Methods for PDEs., ().  doi: 10.1002/num.20552.  Google Scholar [61] D. Yang, A characteristic mixed method with dynamic finite-element space for convection-dominated diffusion problems,, J. Computational and Applied mathematics, 43 (1992), 343.  doi: 10.1016/0377-0427(92)90020-X.  Google Scholar

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##### References:
 [1] T. Arbogast and M. F. Wheeler, A characteristics-mixed finite element method for advection-dominated transport problems,, SIAM J. Numer. Anal., 32 (1995), 404.  doi: 10.1137/0732017.  Google Scholar [2] D. N. Arnolds, L. R. Scott and M. Vogelus, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygonal,, Ann. Scuola. Norm. Sup. Pisa, (1988), 169.   Google Scholar [3] M. Bause and P. Knabner, Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems,, SIAM J. Numer. Anal., 39 (2002), 1954.  doi: 10.1137/S0036142900367478.  Google Scholar [4] J. P. Benque and J. Ronat, Quelques difficulties des modeles numeriques en hydraulique,, Comp. Meth. Appl. Mech. Engrg., (1982), 471.   Google Scholar [5] P. J. Binning and M. A. Celia, A finite volume Eulerian-Lagrangian localized adjoint method for solution of the contaminant transport equations in two-dimensional multi-phase flow systems,, Water Resour. Res., 32 (1996), 103.  doi: 10.1029/95WR02763.  Google Scholar [6] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,, RAIRO Anal. Numér., 8 (1974), 129.   Google Scholar [7] F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,", Springer Series in Computational Mathematics, 15 (1991).   Google Scholar [8] M. A. Celia, T. F. Russell, I. Herrera and R. E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation,, Advances in Water Resources, 13 (1990), 187.  doi: 10.1016/0309-1708(90)90041-2.  Google Scholar [9] Z. Chen, Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems,, Comput. Methods Appl. Mech. Engrg., 191 (2002), 2509.  doi: 10.1016/S0045-7825(01)00411-X.  Google Scholar [10] Z. Chen, S.-H. Chou and D. Y. Kwak, Characteristic-mixed covolume methods for advection-dominated diffusion problems,, Numerical Linear Algebra with Applications, 13 (2006), 677.  doi: 10.1002/nla.492.  Google Scholar [11] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, 4 (1978).  doi: 10.1016/S0168-2024(08)70178-4.  Google Scholar [12] H. K. Dahle, R. E. Ewing and T. F. Russell, Eulerian-Lagrangian localized adjoint methods for a nonlinear convection-diffusion equation,, Comp. Meth. Appl. Mech. Engrg., 122 (1995), 223.  doi: 10.1016/0045-7825(94)00733-4.  Google Scholar [13] C. N. Dawson, T. F. Russell and M. F. Wheeler, Some improved error estimates for the modified method of characteristics,, SIAM J. Numer. Anal., 26 (1989), 1487.  doi: 10.1137/0726087.  Google Scholar [14] J. Douglas Jr., F. Furtado and F. Pereira, On the numerical simulation of water flooding of hetergeneous petroleum reserviors,, Comput. Geosci., 1 (1997), 155.  doi: 10.1023/A:1011565228179.  Google Scholar [15] J. Douglas, Jr., C.-S. Huang and F. Pereira, The modified method of characteristics with adjusted advection,, Numer. Math., 83 (1999), 353.  doi: 10.1007/s002110050453.  Google Scholar [16] J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures,, SIAM J. Numer. Anal., 19 (1982), 871.  doi: 10.1137/0719063.  Google Scholar [17] M. S. Espedal and R. E. Ewing, Characteristic Petrov-Galerkin subdomain methods for two-phase immiscible flow,, Proceedings of the first world congress on computational mechanics (Austin, 64 (1987), 113.   Google Scholar [18] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998).   Google Scholar [19] R. E. Ewing (Ed.), "The Mathematics of Reservoir Simulation,", Research Frontiers in Applied Mathematics 1, (1984).   Google Scholar [20] R. E. Ewing, T. F. Russell and M. F. Wheeler, 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), 73.  doi: 10.1016/0045-7825(84)90048-3.  Google Scholar [21] A. O. Garder, D. W. Peaceman and A. L. Pozzi, Numerical calculations of multidimensional miscible displacement by the method of characteristics,, Soc. Pet. Eng. J., 4 (1964), 26.   Google Scholar [22] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983).   Google Scholar [23] R. W. Healy and T. F. Russell, A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation,, Water Resour. Res., 29 (1993), 2399.  doi: 10.1029/93WR00403.  Google Scholar [24] R. W. Healy and T. F. Russell, Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method,, Adv. Water Res., 21 (1998), 11.   Google Scholar [25] J. M. Hervouet, Applications of the method of characteristics in their weak formulation to solving two-dimensional advection-equations on mesh grids,, in, (1986), 149.   Google Scholar [26] C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems,, RAIRO Anal. Numer., 15 (1981), 41.   Google Scholar [27] X. Li, W. Wu and O. C. Zienkiewicz, Implicit characteristic Galerkin method for convection-diffusion equations,, Int. J. Numer. Meth. Engrg., 47 (2000), 1689.  doi: 10.1002/(SICI)1097-0207(20000410)47:10<1689::AID-NME850>3.0.CO;2-W.  Google Scholar [28] K. W. Morton, A. Priestley and E. Süli, Stability of the Lagrangian-Galerkin method with nonexact integration,, RAIRO Model. Math. Anal. Num., 22 (1988), 625.   Google Scholar [29] J. C. Nédélec, A new family of mixed finite elements in $\mathbf R^3$,, Numerische Mathematik, 50 (1986), 57.  doi: 10.1007/BF01389668.  Google Scholar [30] S. P. Neuman, An Eulerian-Lagrangian numerical scheme for the dispersion-convection equation using conjugate space-time grids,, J. Comp. Phys., 41 (1981), 270.  doi: 10.1016/0021-9991(81)90097-8.  Google Scholar [31] D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation,", Elsevier, (1977).   Google Scholar [32] G. F. Pinder and H. H. Cooper, A numerical technique for calculating the transient position of the saltwater front,, Water Resou. Res., (1970), 875.   Google Scholar [33] O. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations,, Numer. Math., 38 (): 309.  doi: 10.1007/BF01396435.  Google Scholar [34] P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems,, Mathematical Aspects of the Finite Element Method, 606 (1975), 292.   Google Scholar [35] H.-G. Roos, M. Stynes and L. Tobiska, "Numerical Methods for Singularly Perturbed Differential Equations,", Convection-Diffusion and Flow Problems, (1996).   Google Scholar [36] E. Varoglu and W. D. L. Finn, Finite elements incorporating characteristics for one-dimensional diffusion-convection equation,, J. Comput. Phys., 34 (1980), 371.  doi: 10.1016/0021-9991(80)90095-9.  Google Scholar [37] H. Wang, A family of ELLAM schemes for advection-diffusion-reaction equations and their convergence analyses,, Numerical Methods for PDEs, 14 (1998), 739.   Google Scholar [38] H. Wang, An optimal-order error estimate for an ELLAM scheme for two-dimensional linear advection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1338.  doi: 10.1137/S0036142998335686.  Google Scholar [39] H. Wang, An optimal-order error estimate for MMOC and MMOCAA schemes for multidimensional advection-reaction equations,, Numerical Methods for PDEs, 18 (2002), 69.   Google Scholar [40] H. Wang, An optimal-order error estimate for a family of ELLAM-MFEM approximations to porous medium flow,, SIAM J. Numer. Anal., 46 (2008), 2133.  doi: 10.1137/S0036142903428281.  Google Scholar [41] H. Wang and M. Al-Lawatia, A locally conservative Eulerian-Lagrangian control-volume method for transient advection-diffusion equations,, Numerical Methods for Partial Differential Equations, 22 (2005), 577.  doi: 10.1002/num.20106.  Google Scholar [42] H. Wang, H. K. Dahle, R. E. Ewing, M. S. Espedal, R. C. Sharpley and S. Man, An ELLAM scheme for advection-diffusion equations in two dimensions,, SIAM J. Sci. Comput., 20 (1999), 2160.  doi: 10.1137/S1064827596309396.  Google Scholar [43] H. Wang, R. E. Ewing, G. Qin and S. L. Lyons, "An Eulerian-Lagrangian Formulation for Compositional Flow in Porous Media,", The 2006 Society of Petroleum Engineering Annual Technical Conference in San Antonio, (2006), 24.   Google Scholar [44] H. Wang, R. E. Ewing, G. Qin, S. L. Lyons, M. Al-Lawatia and S. Man, A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations,, J. Comput. Phys., 152 (1999), 120.  doi: 10.1006/jcph.1999.6239.  Google Scholar [45] H. Wang, R. E. Ewing and T. F. Russell, Eulerian-Lagrangian localized methods for convection-diffusion equations and their convergence analysis,, IMA J. Numer. Anal., 15 (1995), 405.  doi: 10.1093/imanum/15.3.405.  Google Scholar [46] H. Wang, X. Shi and R. E. Ewing, An ELLAM scheme for multidimensional advection-reaction equations and its optimal-order error estimate,, SIAM. J. Numer. Anal., 38 (2001), 1846.  doi: 10.1137/S0036142999362389.  Google Scholar [47] H. Wang and K. Wang, Uniform estimates for Eulerian-Lagrangian methods for singularly perturbed time-dependent problems,, SIAM J. Numer. Anal., 45 (2007), 1305.  doi: 10.1137/060652816.  Google Scholar [48] K. Wang, A uniformly optimal-order error estimate of an ELLAM scheme for unstady-state advection-diffusion equations,, International Journal of Numerical Analysis and Modeling, 5 (2008), 286.   Google Scholar [49] K. Wang, An optimal-order estimate for MMOC-MFEM approximations to porous medium flow,, Numer. Methods for Partial Differential Equations, 25 (2008), 1283.  doi: 10.1002/num.20397.  Google Scholar [50] K. Wang, A uniform optimal-order estimate for an Eulerian-Lagrangian discontinuous Galerkin method for transient advection-diffusion equations,, Numer. Methods for Partial Differential Equations, 25 (2009), 87.  doi: 10.1002/num.20338.  Google Scholar [51] K. Wang and H. Wang, A uniform estimate for the ELLAM scheme for transport equations,, Numer. Methods for PDEs, 24 (2008), 535.   Google Scholar [52] K. Wang and H. Wang, An optimal-order error estimate to the modified method of characteristics for a degenerate convection-diffusion equation,, International Journal of Numerical Analysis and Modeling, 6 (2009), 217.   Google Scholar [53] K. Wang and H. Wang, A uniform estimate for the MMOC for two-dimensional advection-diffusion equations,, Numer. Methods for PDEs, 26 (2010), 1054.   Google Scholar [54] K. Wang, H. Wang and M. Al-Lawatia, An Eulerian-Lagrangian discontinuous Galerkin method for transient advection-diffusion equations,, Numer. Methods for Partial Differential Equations, 23 (2007), 1343.  doi: 10.1002/num.20223.  Google Scholar [55] K. Wang, H. Wang and M. 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