Free access
Issue
ESAIM: M2AN
Volume 43, Number 5, September-October 2009
Page(s) 957 - 971
DOI http://dx.doi.org/10.1051/m2an/2009017
Published online 12 June 2009
  1. R. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, New York (2003).
  2. C. Bi and V. Ginting, Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 107 (2007) 177–198. [CrossRef]
  3. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 2nd edn., (2002).
  4. Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713–735. [CrossRef] [MathSciNet]
  5. Z. Cai, J. Mandel and S. McCormick, The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392–402. [CrossRef] [MathSciNet]
  6. C. Carstensen, R. Lazarov and S. Tomov, Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005) 2496–2521. [CrossRef] [MathSciNet]
  7. P. Chatzipantelidis and R.D. Lazarov, Error estimates for a finite volume element method for elliptic PDEs in nonconvex polygonal domains. SIAM J. Numer. Anal. 42 (2004) 1932–1958. [CrossRef]
  8. P. Chatzipantelidis, R.D. Lazarov and V. Thomée, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer. Meth. PDEs 20 (2004) 650–674.
  9. S.H. Chou and D.Y. Kwak, Multigrid algorithms for a vertex-centered covolume method for elliptic problems. Numer. Math. 90 (2002) 459–486. [CrossRef] [MathSciNet]
  10. S.H. Chou and Q. Li, Error estimates in L2, H1 and Formula in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69 (2000) 103–120. [CrossRef] [MathSciNet]
  11. S.H. Chou, D.Y. Kwak and Q. Li, Lp error estimates and superconvergence for covolume or finite volume element methods. Numer. Meth. PDEs 19 (2003) 463–486.
  12. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
  13. C.N. Dawson, M.F. Wheeler and C.S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35 (1998) 435–452. [CrossRef] [MathSciNet]
  14. J. de Frutos and J. Novo, Postprocessing the linear finite element method. SIAM J. Numer. Anal. 40 (2002) 805–819. [CrossRef] [MathSciNet]
  15. R.E. Ewing, T. Lin and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865–1888. [CrossRef] [MathSciNet]
  16. R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods: Handbook of Numerical Analysis. North-Holland, Amsterdam (2000).
  17. M. Feistauer, J. Felcman, M. Lukáčová-Medvidová and G. Warnecke, Error estimates of a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 1528–1548. [CrossRef] [MathSciNet]
  18. B. García-Archilla, J. Novo and E.S. Titi, Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds. SIAM J. Numer. Anal. 35 (1998) 941–972. [CrossRef] [MathSciNet]
  19. B. García-Archilla and E.S. Titi, Postprocessing the Galerkin method: the finite element case. SIAM J. Numer. Anal. 37 (2000) 470–499. [CrossRef] [MathSciNet]
  20. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840. Springer-Verlag, New York (1989).
  21. A. Lasis and E. Süli, hp-version discontinuous Galerkin finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 45 (2007) 1544–1569. [CrossRef] [MathSciNet]
  22. R. Li, Z. Chen and W. Wu, Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000).
  23. X. Ma, S. Shu and A. Zhou, Symmetric finite volume discretizations for parabolic problems. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4467–4485. [CrossRef] [MathSciNet]
  24. M. Marion and J.C. Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32 (1995) 1170–1184. [CrossRef] [MathSciNet]
  25. H. Rui, Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems. J. Comput. Appl. Math. 146 (2002) 373–386. [CrossRef] [MathSciNet]
  26. A.H. Schatz, V. Thomée and L. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265–304. [CrossRef] [MathSciNet]
  27. R.K. Sinha and J. Geiser, Error estimates for finite volume element methods for convection-diffusion-reaction equations. Appl. Numer. Math. 57 (2007) 59–72. [CrossRef] [MathSciNet]
  28. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68. Springer-Verlag, Berlin (1988).
  29. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997).
  30. V. Thomée and L. Wahlbin, On Galerkin methods in semilinear parabolic problems. SIAM J. Numer. Anal. 12 (1975) 378–389. [CrossRef] [MathSciNet]
  31. Y. Yan, Postprocessing the finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 44 (2006) 1681–1702. [CrossRef] [MathSciNet]
  32. M. Yang, A second-order finite volume element method on quadrilateral meshes for elliptic equations. ESAIM: M2AN 40 (2006) 1053–1067. [CrossRef] [EDP Sciences]
  33. X. Ye, A discontinuous finite volume method for the Stokes problems. SIAM J. Numer. Anal. 44 (2006) 183–198. [CrossRef] [MathSciNet]
  34. S. Zhang, On domain decomposition algorithms for covolume methods for elliptic problems. Comput. Methods Appl. Mech. Engrg. 196 (2006) 24–32. [CrossRef] [MathSciNet]

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