160
Views
14
CrossRef citations to date
0
Altmetric
Section B

A full discrete two-grid finite-volume method for a nonlinear parabolic problem

, &
Pages 1644-1663 | Received 17 May 2009, Accepted 27 Aug 2010, Published online: 14 Mar 2011

References

  • Adams , R. 1975 . Sobolev Spaces , New York : Academic Press .
  • Bank , R. and Rose , D. 1987 . Some error estimates for the box method . SIAM J. Numer. Anal. , 24 ( 4 ) : 777 – 787 .
  • Barrett , J. , Blowey , J. and Garcke , H. 1998 . Finite element approximation of a fourth order nonlinear degenerate parabolic equation . Numer. Math. , 80 ( 4 ) : 525 – 556 .
  • Bi , C. and Ginting , V. 2007 . Two-grid finite volume element method for linear and nonlinear elliptic problems . Numer. Math. , 108 ( 2 ) : 177 – 198 .
  • Brenner , S. and Scott , L. 1994 . The Mathematics Theory of Finite Element Methods , New York : Springer-Verlag .
  • Cai , Z. 1991 . On the finite volume method . Numer. Math. , 58 ( 7 ) : 713 – 735 .
  • Cai , Z. , Mandel , J. and McCormick , S. 1991 . The finite volume element method for diffusion equations on general triangulations . SIAM J. Numer. Anal. , 28 ( 2 ) : 392 – 402 .
  • Chen , Y. , Yang , M. and Bi , C. 2009 . Two-grid methods for finite volume element approximations of nonlinear parabolic equations . J. Comput. Appl. Math. , 228 ( 1 ) : 123 – 132 .
  • Chou , S. and Li , Q. 2000 . Error estimates in L 2, H 1 and L ∞ in covolume methods for elliptic and parabolic problem: A unified approach . Math. Comput. , 69 ( 229 ) : 103 – 120 .
  • Dawson , C. and Wheeler , M. 1994 . Two-grid methods for mixed finite element approximations of nonlinear parabolic equations . Contemp. Math. , 180 : 191 – 203 .
  • Dawson , C. , Wheeler , M. and Woodward , C. 1998 . A two-grid finite difference scheme for nonlinear parabolic equations . SIAM J. Numer. Anal. , 35 ( 2 ) : 435 – 452 .
  • Dolejsi , V. , Feistauer , M. and Schwab , C. 2002 . A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems . CALCOLO , 39 ( 1 ) : 1 – 40 .
  • Drblikova , O. and Mikula , K. 2008 . Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing . SIAM J. Numer. Anal. , 46 ( 1 ) : 37 – 60 .
  • Ewing , R. , Iliev , O. and Lazarov , R. 2001 . A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients . SIAM J. Sci. Comput. , 23 ( 4 ) : 1334 – 1350 .
  • Ewing , R. , Lin , T. and Lin , Y. 2002 . On the accuracy of the finite volume element based on piecewise linear polynomials . SIAM J. Numer. Anal. , 39 ( 6 ) : 1865 – 1888 .
  • Eymard , R. , Gallouet , T. and Herbin , R. 1997 . “ Finite volume methods ” . In Handbook of Numerical Analysis , Edited by: Ciarlet , P. G. and Lions , J. L. Vol. VII , 713 – 1020 . Elsevier Science B.V., North-Holland .
  • Feistauer , M. , Felcman , J. Lukacova-Medvid'ova , M. 1999 . Error estimates for a combined finite volume–finite element method for nonlinear convection–diffusion problems . SIAM J. Numer. Anal. , 36 ( 5 ) : 1528 – 1548 .
  • Gao , F. 2005 . Finite volume element predictor–corrector method for a class of nonlinear parabolic systems . Northeast Math. J. , 21 ( 3 ) : 305 – 314 .
  • Gao , F. and Yuan , Y. 2008 . The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem . Comput. Math. Appl. , 56 ( 1 ) : 71 – 81 .
  • He , G. , He , Y. and Feng , X. 2007 . Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem . Numer. Methods Partial Differential Equations , 23 ( 5 ) : 1167 – 1191 .
  • Jones , J. , Cai , Z. , McCormick , S. and Russell , T. 1997 . Control-volume mixed finite element methods . Comput. Geosci. , 1 ( 3 ) : 289 – 315 .
  • Kriva , Z. and Mikula , K. 2002 . An adaptive finite volume scheme for solving nonlinear diffusion equations in image processing . J. Vis. Commun. Image Represent. , 13 ( 1 ) : 22 – 35 .
  • Layton , W. and Lenferink , W. 1995 . Two-level Picard and modified Picard methods for the Navier-Stokes equations . Appl. Math. Comp. , 69 ( 1 ) : 263 – 274 .
  • Li , R. 1987 . Generalized difference methods for a nonlinear Dirichlet problem . SIAM J. Numer. Anal. , 24 ( 1 ) : 77 – 88 .
  • Li , R. , Chen , Z. and Wu , W. 2000 . Generalized Difference Methods for Differential Equations , New York : Marcel Dekker, Inc .
  • Marion , M. and Xu , J. 1995 . Error estimates on a new nonlinear Galerkin method based on two-grid finite elements . SIAM J. Numer. Anal. , 32 : 1170 – 1184 .
  • Mikula , K. and Ramarosy , N. 2001 . Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing . Numer. Math. , 89 ( 3 ) : 561 – 590 .
  • Morton , K. 1996 . “ Numerical solution of convection–diffusion problems ” . London : Chapman & Hall .
  • Rachford , H. 1973 . Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations . SIAM J. Numer. Anal. , 10 ( 6 ) : 1010 – 1026 .
  • Russell , T. 1985 . Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media . SIAM J. Numer. Anal. , 22 ( 5 ) : 970 – 1013 .
  • Thomée , V. 1997 . Galerkin Finite Element Method for Parabolic Problems , Berlin : Springer-Verlag .
  • Wu , H. and Li , R. 2003 . Error estimates for finite volume element methods for general second order elliptic problem . Numer. Methods Partial Differential Equations , 19 : 693 – 708 .
  • Xu , J. 1994 . A novel two-grid method for semi-linear elliptic equations . SIAM J. Sci. Comput. , 15 ( 1 ) : 231 – 237 .
  • Xu , J. 1996 . Two-grid discretization techniques for linear and nonlinear PDEs . SIAM J. Numer. Anal. , 33 ( 5 ) : 1759 – 1777 .
  • Xu , J. and Zou , Q. 2009 . Analysis of linear and quadratic simplicial finite volume methods for elliptic equations . Numer. Math. , 111 ( 3 ) : 469 – 492 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.