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A Finite Volume Element Method for a Nonlinear Parabolic Problem

  • P. ChatzipantelidisEmail author
  • V. Ginting
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 45)

Abstract

We study a finite volume element discretization of a nonlinear parabolic equation in a convex polygonal domain. We show the existence of the discrete solution and derive error estimates in L 2- and H 1-norms. We also consider a linearized method and provide numerical results to illustrate our theoretical findings.

Keywords

Nonlinear parabolic problem Finite volume element method Error estimates 

Notes

Acknowledgements

The research of P. Chatzipantelidis was partly supported by the FP7-REGPOT-2009-1 project “Archimedes Center for Modeling Analysis and Computation,” funded by the European Commission. The research of V. Ginting was partially supported by the grants from DOE (DE-FE0004832 and DE-SC0004982), the Center for Fundamentals of Subsurface Flow of the School of Energy Resources of the University of Wyoming (WYDEQ49811GNTG, WYDEQ49811PER), and from NSF (DMS-1016283).

References

  1. 1.
    Chatzipantelidis, P., Ginting, V., Lazarov, R.D.: A finite volume element method for a non-linear elliptic problem. Numer. Linear Algebra Appl. 12, 515–546 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer. Methods Partial Differ. Equ. 20, 650–674 (2004)zbMATHCrossRefGoogle Scholar
  3. 3.
    Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation. Elsevier Science Publisher, B.V. Amsterdam, (1986)zbMATHGoogle Scholar
  4. 4.
    Chou, S.-H., Li, Q.: Error estimates in L 2, H 1n and L in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69, 103–120 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)Google Scholar
  6. 6.
    Keller, E., Segel, L.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefGoogle Scholar
  7. 7.
    Ladyženskaja, O.A., Solonnikov, V.A., Uraĺceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. American Mathematical Society, Providence (1968)Google Scholar
  8. 8.
    Li, R.: Generalized difference methods for a nonlinear Dirichlet problem. SIAM J. Numer. Anal. 24, 77–88 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Li, R., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations. Marcel Dekker, New York (2000)zbMATHGoogle Scholar
  10. 10.
    Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)zbMATHCrossRefGoogle Scholar
  11. 11.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)zbMATHGoogle Scholar
  12. 12.
    Zhang, T., Zhong, H., Zhao, J.: A fully discrete two–grid finite–volume method for a nonlinear parabolic problem. Int. J. Comput. Math. 88, 1644–1663 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CreteHeraklionGreece
  2. 2.Department of MathematicsUniversity of WyomingLaramieUSA

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