Journal of Scientific Computing

, Volume 77, Issue 3, pp 1660–1678 | Cite as

Error Analysis of Mixed Finite Element Methods for Nonlinear Parabolic Equations

  • Huadong Gao
  • Weifeng QiuEmail author


In this paper, we prove a discrete embedding inequality for the Raviart–Thomas mixed finite element methods for second order elliptic equations, which is analogous to the Sobolev embedding inequality in the continuous setting. Then, by using the proved discrete embedding inequality, we provide an optimal error estimate for linearized mixed finite element methods for nonlinear parabolic equations. Several numerical examples are provided to confirm the theoretical analysis.


Nonlinear parabolic equations Finite element method Discrete Sobolev embedding inequality Unconditional convergence Optimal error analysis 

Mathematics Subject Classification

35Q30 65M60 65N30 



The authors would like to thank Prof. Weiwei Sun for useful discussions.


  1. 1.
    Akrivis, G., Li, B., Lubich, C.: Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp. 86, 1527–1552 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arbogast, T., Estep, D., Sheehan, B., Tavener, S.: A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary. SIAM/ASA J. Uncertain. Quantif. 3, 169–198 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arbogast, T., Wheeler, M.: A characteristics-mixed finite element method for advection-dominated transport problems. SIAM J. Numer. Anal. 32, 404–424 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Brezzi, F., Marini, L., Micheletti, S., Pietra, P., Sacco, R., Wang, S.: Discretization of semiconductor device problems (I), Handbook of Numerical Analysis XIII, special Volume on Numerical Methods in Electromagnetics, pp. 317–442. North-Holland, Amsterdam (2005)zbMATHGoogle Scholar
  6. 6.
    Buffa, A., Ortner, C.: Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 29, 827–855 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, L., Chen, Y.: Two-grid method for nonlinear reaction–diffusion equations by mixed finite element methods. J. Sci. Comput. 49, 383–401 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, Y., Liu, H., Liu, S.: Analysis of two-grid methods for reaction–diffusion equations by expanded mixed finite element methods. Int. J. Numer. Meth. Eng. 69, 408–422 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, Y., Huang, Y., Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction–diffusion equations. Int. J. Numer. Meth. Eng. 57, 193–209 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Di Pietro, D., Ern, A.: Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations. Math. Comput. 79, 1303–1330 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dawson, C., Sun, S., Wheeler, M.: Compatible algorithm for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193, 2562–2580 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Egger, H., Schoberl, J.: A hybrid mixed discontinuous Galerkin finite-element method for convection–diffusion problems. IMA J. Numer. Anal. 30, 1206–1234 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ern, A., Guermond, J.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefGoogle Scholar
  14. 14.
    Ewing, R., Wheeler, M.: Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17, 351–365 (1980)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Feng, X., He, Y., Liu, C.: Analysis of finite element approximations of a phase field model for two phase fluids. Math. Comp. 76, 539–571 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Feng, X., Wu, H.: A posteriori error estimates and an adaptive finite element method for the Allen–Cahn equation and the mean curvature flow. J. Sci. Comput. 24, 121–146 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gadau, S., Jungel, A.: A three-dimensional mixed finite-element approximation of the semiconductor energy-transport equations. SIAM J. Sci. Comput. 31, 1120–1140 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gao, H., Li, B., Sun, W.: Optimal error estimates of linearized Crank–Nicolson Galerkin FEMs for the time-dependent Ginzburg–Landau equations in superconductivity. SIAM J. Numer. Anal. 52, 1183–1202 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gao, H., Li, B., Sun, W.: Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon. Numer. Math. 136, 383–409 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Garcia, M.: Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: the continuously-time case. Numer. Methods Partial Differ. Equ. 10, 129–149 (1994)CrossRefGoogle Scholar
  21. 21.
    Garcia, M.: Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: the discrete-time case. Numer. Methods Partial Differ. Equ. 10, 149–169 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Holst, S., Jungel, A., Pietra, P.: An adaptive mixed scheme for energy-transport simulations of field-effect transistors. SIAM J. Sci. Comput. 25, 1698–1716 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hou, Y., Li, B., Sun, W.: Error estimates of splitting Galerkin methods for heat and sweat transport in textile materials. SIAM J. Numer. Anal. 51, 88–111 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hua, J., Lin, P., Liu, C., Wang, Q.: Energy law preserving \(C^0\) finite element schemes for phase field models in two-phase flow computations. J. Comput. Phys. 230, 7115–7131 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Johnson, C., Thomee, V.: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numer. 15, 41–78 (1981)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kim, D., Park, E., Seo, B.: Two-scale product approximation for semilinear parabolic problems in mixed methods. J. Korean Math. Soc. 51, 267–288 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Logg, A., Mardal, K., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)CrossRefGoogle Scholar
  29. 29.
    Li, B.: Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp. (2017). MathSciNetCrossRefGoogle Scholar
  30. 30.
    Li, B., Gao, H., Sun, W.: Unconditionally optimal error estimates of a Crank–Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52, 933–954 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Li, B., 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, 1959–1977 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Li, B., Sun, W.: Regularity of the diffusion–dispersion tensor and error analysis of FEMs for a porous media flow. SIAM J. Numer. Anal. 53, 1418–1437 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Nedelec, J.: Mixed finite elements in \(\mathbf{R}^3\). Numer. Math. 35, 315–341 (1980)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa 20(4), 733–737 (1966)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. In: Mathematical Aspects of the Finite Element Method. Lecture Notes in Math, vol. 606. Springer, New York (1977)Google Scholar
  37. 37.
    Wu, L., Allen, M.: A two-grid method for mixed finite-element solution of reaction–diffusion equations. Numer. Methods Partial Differ. Equ. 15, 317–332 (1999)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  2. 2.Hubei Key Laboratory of Engineering Modeling and Scientific ComputingHuazhong University of Science and TechnologyWuhanChina
  3. 3.Department of MathematicsCity University of Hong KongKowloonHong Kong

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