Journal of Scientific Computing

, Volume 58, Issue 1, pp 90–114 | Cite as

On Adaptive Eulerian–Lagrangian Method for Linear Convection–Diffusion Problems

  • Xiaozhe Hu
  • Young-Ju Lee
  • Jinchao Xu
  • Chen-Song Zhang


In this paper, we consider the adaptive Eulerian–Lagrangian method (ELM) for linear convection–diffusion problems. Unlike classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity. Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate the efficiency and robustness of our adaptive algorithm.


Convection–diffusion problems Eulerian–Lagrangian method A posteriori error estimation Adaptive mesh refinement 

Mathematics Subject Classification (2000)

65M60 65M25 65M15 



The authors would like to thank Professor Ricardo H. Nochetto, Professor Long Chen, and two anonymous referees for their comments on earlier versions of this paper. Hu and Xu are supported in part by NSF Grant DMS-0915153 and DOE Grant DE-SC0006903. Zhang is partially supported by the Dean Startup Fund, Academy of Mathematics and System Sciences, NSFC-91130011, and State High Tech Development Plan of China (863 Program) 2012AA01A309.


  1. 1.
    Abbott, M.B.: An Introduction to Method of Characteristics. American Elsevier, New York (1966)Google Scholar
  2. 2.
    Achdou, Y., Guermond, J.L.: Convergence analysis of a finite element projection/Lagrange–Galerkin method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 37(3), 799–826 (2000)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Adjerid, S., Belguendouz, B., Flaherty, J.E.: A posteriori finite element error estimation for diffusion problems. SIAM J. Sci. Comput. 21(2), 728–746 (1999)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [Wiley], New York (2000)CrossRefGoogle Scholar
  5. 5.
    Bernardi, C., Verfürth, R.: A posteriori error analysis of the fully discretized time-dependent Stokes equations. M2AN. Math. Model. Numer. Anal. 38(3), 437–455 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Binev, P., Dahmen, W., Devore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Brooks, N., Hughes, T.J.: Streamline Upwind/Petrov–Galerkink formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cawood, M.E., Ervin, V.J., Layton, W.J., Maubach, J.M.: Adaptive defect correction methods for convection dominated, convection diffusion problems. J. Comput. Appl. Math. 116(1), 1–21 (2000)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Chen, Z., Feng, J.: An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comput. 73(247), 1167–1193 (2004)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Chen, Z., Ji, G.: Adaptive computation for convection dominated diffusion problems. Sci. China Ser. A 47(suppl), 22–31 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Chen, Z., Ji, G.: Sharp \(L^1\) a posteriori error analysis for nonlinear convection–diffusion problems. Math. Comput. 75(253), 43–71 (2006)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Chen, Z., Nochetto, R.H., Schmidt, A.: A characteristic galerkin method with adaptive error control for the continuous casting problem. Comput. Methods Appl. Mech. Eng. 189(1), 249–276 (2000)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Davis, GdV, Mallinson, G.D.: An evaluation of upwind and central difference approximations by a study of recirculating flow. Comput. Fluids 4, 29–43 (1976)CrossRefMATHGoogle Scholar
  16. 16.
    Demkowicz, L., Oden, J.T.: An adaptive characteristic Petrov–Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables. Comput. Methods Appl. Mech. Eng. 55(1–2), 63–87 (1986)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Douglas Jr, J., Russell, T.F.: 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(5), 871–885 (1982)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems II: optimal error estimates in \(l_\infty l_2 \) and \(l_\infty l_\infty \). SIAM J. Numer. Anal. 32(3), 706–740 (1995a)Google Scholar
  21. 21.
    Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems IV: nonlinear problems. SIAM J. Numer. Anal. 32(6), 1729–1749 (1995b)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Etienne, J., Hinch, E., Li, J.: A Lagrangian–Eulerian approach for the numerical simulation of freesurface flow of a viscoelastic material. J. Non-Newton. Fluid Mech. 136, 157–166 (2006)CrossRefMATHGoogle Scholar
  23. 23.
    Ewing, R.E.: The Mathematics of Reservoir Simulation. SIAM, Philadelphia (1983)CrossRefMATHGoogle Scholar
  24. 24.
    Feng, K., Shang, Zj: Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 463, 451–463 (1995)MathSciNetGoogle Scholar
  25. 25.
    Hebeker, F.K., Rannacher, R.: An adaptive finite element method for unsteady convection-dominated flows with stiff source terms. SIAM J. Sci. Comput. 21(3), 799–818 (1999)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Houston, P., Süli, E.: Adaptive Lagrange–Galerkin methods for unsteady convection–diffusion problems. Math. Comput. 70(233), 77–106 (2001)CrossRefMATHGoogle Scholar
  27. 27.
    Jakob, M.: Heat Transfer. Wiley, New York (1959)Google Scholar
  28. 28.
    Jia, J., Hu, X., Xu, J., Zhang, C.S.: Effects of integrations and adaptivity for the Eulerian–Lagrangian method. J. Comput. Math. 29, 367–395 (2011)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Johnson, C., Nie, Y.Y., Thomée, V.: An a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem. SIAM J. Numer. Anal. 27(2), 277–291 (1990)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Karakatsani, F., Makridakis, C.: A posteriori estimates for approximations of time-dependent Stokes equations. IMA J. Numer. Anal. 27(4), 741–764 (2007)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Kharrat, N., Mghazli, Z.: Residual error estimators for the time-dependent Stokes equations. Comptes Rendus Mathe. 340(5), 405–408 (2005)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Krüger, O., Picasso, M., Scheid, J.F.: A posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification. Comput. Methods Appl. Mech. Eng. 192(5–6), 535–558 (2003)CrossRefMATHGoogle Scholar
  33. 33.
    Lakkis, O., Makridakis, C.: Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 75(256), 1627–1658 (2006)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Lee, Y., Xu, J.: New formulations, positivity preserving discretizations and stability analysis for non-newtonian flow models. Comput. Methods Appl. Mech. Eng. 195, 1180–1206 (2006)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Moore, P.K.: A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension. SIAM J. Numer. Anal. 31(1), 149–169 (1994)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Nochetto, R.H., Savaré, G., Verdi, C.: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math. 53(5), 525–589 (2000)CrossRefMATHGoogle Scholar
  38. 38.
    Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: An introduction. In: Multiscale, Nonlinear and Adaptive Approximation. Springer, Berlin, Heidelberg pp. 409–542 (2009)Google Scholar
  39. 39.
    Phillips, T.N., Williams, A.J.: Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method. J. Non-Newton. Fluid Mech. 87(2–3), 215–246 (1999)CrossRefMATHGoogle Scholar
  40. 40.
    Phillips, T.N., Williams, A.J.: A semi-Lagrangian finite volume method for Newtonian contraction flows. SIAM J. Sci. Comput. 22(6), 2152–2177 (2000)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Picasso, M.: Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Eng. 167(3–4), 223–237 (1998)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math. 38(3), 309–332 (1982)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Roos, H., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection–Diffusion–Reaction and Flow Problems. Springer, Berlin Heidelberg (2008)Google Scholar
  44. 44.
    Schmidt, A., Siebert, K.G.: Design of adaptive finite element software: the finite element toolbox ALBERTA. In: Lecture Notes in Computational Science and Engineering, vol. 42. Springer, Berlin (2005)Google Scholar
  45. 45.
    Sorbents, I., Totsche, K.U., Knabner, P., Kgel-knabner, I.: The modeling of reactive solute transport with sorption to mobile and immobile sorbents, 1. Experimental evidence and model development. Water Resour. Res. 32, 1611–1622 (1996)CrossRefGoogle Scholar
  46. 46.
    Stevenson, R.: An optimal adaptive finite element method. SIAM J. Numer. Anal. 42(5), 2188–2217 (2005)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh refinement techniques. Wiley and Teubner, New York (1996)MATHGoogle Scholar
  48. 48.
    Verfürth, R.: A posteriori error estimates for nonlinear problems: \(L^r(0,T;W^{1,\rho }(\Omega ))\)-error estimates for finite element discretizations of parabolic equations. Numer. Methods Partial Diff. Equ. 14(4), 487–518 (1998)CrossRefMATHGoogle Scholar
  49. 49.
    Verfurth, R.: Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal. 43, 1783–1802 (2005)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Wang, H.: An optimal-order error estimate for an ELLAM scheme for two-dimensional linear advection–diffusion equations. SIAM J. Numer. Anal. 37(4), 1338–1368 (2000)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Wang, H., Wang, K.: Uniform estimates for Eulerian–Lagrangian methods for singularly perturbed time-dependent problems. SIAM J. Numer. Anal. 45(3), 1305–1329 (2007)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Xiu, D., Karniadakis, G.E.: A semi-Lagrangian high-order method for Navier–Stokes equations. J. Comput. Phys. 172(2), 658–684 (2001)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Yeh, G.T.: An adaptive local grid refinement based on the exact peak capture and oscillation free scheme to solve transport equations. Comput. Fluid 24, 293 (1995)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Xiaozhe Hu
    • 2
    • 1
  • Young-Ju Lee
    • 3
  • Jinchao Xu
    • 1
  • Chen-Song Zhang
    • 4
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Engineering MechanicsKunming University of Science and TechnologyKunmingChina
  3. 3.Department of the MathematicsRutgers, The State University of New JerseyPiscatawayUSA
  4. 4.NCMIS and LSECAcademy of Mathematics and System SciencesBeijingChina

Personalised recommendations