Lagrangian and moving mesh methods for the convection diffusion equation

Konstantinos Chrysafinos; Noel J. Walkington

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 1, page 25-55
  • ISSN: 0764-583X

Abstract

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We propose and analyze a semi Lagrangian method for the convection-diffusion equation. Error estimates for both semi and fully discrete finite element approximations are obtained for convection dominated flows. The estimates are posed in terms of the projections constructed in [Chrysafinos and Walkington, SIAM J. Numer. Anal. 43 (2006) 2478–2499; Chrysafinos and Walkington, SIAM J. Numer. Anal. 44 (2006) 349–366] and the dependence of various constants upon the diffusion parameter is characterized. Error estimates independent of the diffusion constant are obtained when the velocity field is computed exactly.

How to cite

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Chrysafinos, Konstantinos, and Walkington, Noel J.. "Lagrangian and moving mesh methods for the convection diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 25-55. <http://eudml.org/doc/250339>.

@article{Chrysafinos2008,
abstract = { We propose and analyze a semi Lagrangian method for the convection-diffusion equation. Error estimates for both semi and fully discrete finite element approximations are obtained for convection dominated flows. The estimates are posed in terms of the projections constructed in [Chrysafinos and Walkington, SIAM J. Numer. Anal. 43 (2006) 2478–2499; Chrysafinos and Walkington, SIAM J. Numer. Anal. 44 (2006) 349–366] and the dependence of various constants upon the diffusion parameter is characterized. Error estimates independent of the diffusion constant are obtained when the velocity field is computed exactly. },
author = {Chrysafinos, Konstantinos, Walkington, Noel J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Convection diffusion; moving meshes; Lagrangian formulation.; Lagrangian formulation; convection-diffusion equation; error estimates; finite element; convection dominated flows},
language = {eng},
month = {1},
number = {1},
pages = {25-55},
publisher = {EDP Sciences},
title = {Lagrangian and moving mesh methods for the convection diffusion equation},
url = {http://eudml.org/doc/250339},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Chrysafinos, Konstantinos
AU - Walkington, Noel J.
TI - Lagrangian and moving mesh methods for the convection diffusion equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/1//
PB - EDP Sciences
VL - 42
IS - 1
SP - 25
EP - 55
AB - We propose and analyze a semi Lagrangian method for the convection-diffusion equation. Error estimates for both semi and fully discrete finite element approximations are obtained for convection dominated flows. The estimates are posed in terms of the projections constructed in [Chrysafinos and Walkington, SIAM J. Numer. Anal. 43 (2006) 2478–2499; Chrysafinos and Walkington, SIAM J. Numer. Anal. 44 (2006) 349–366] and the dependence of various constants upon the diffusion parameter is characterized. Error estimates independent of the diffusion constant are obtained when the velocity field is computed exactly.
LA - eng
KW - Convection diffusion; moving meshes; Lagrangian formulation.; Lagrangian formulation; convection-diffusion equation; error estimates; finite element; convection dominated flows
UR - http://eudml.org/doc/250339
ER -

References

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  1. R. Balasubramaniam and K. Mutsuto, Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Numer. Methods Fluids7 (1987) 953–984.  
  2. R.E. Bank and R.F. Santos, Analysis of some moving space-time finite element methods. SIAM J. Numer. Anal. 30 (1993) 1–18.  
  3. M. Bause and P. Knabner, Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems. SIAM J. Numer. Anal. 39 (2002) 1954–1984 (electronic).  
  4. J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L2 projection in H1(Ω). Math. Comp. 71 (2002) 147–156 (electronic).  
  5. N.N. Carlson and K. Miller, Design and application of a gradient-weighted moving finite element code. II. In two dimensions. SIAM J. Sci. Comput. 19 (1998) 766–798 (electronic).  
  6. C. Carstensen, Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H1-stability of the L2-projection onto finite element spaces. Math. Comp. 71 (2002) 157–163 (electronic).  
  7. K. Chrysafinos and J.N. Walkington, Error estimates for the discontinuous Galerkin methods for implicit parabolic equations. SIAM J. Numer. Anal. 43 (2006) 2478–2499.  
  8. K. Chrysafinos and J.N. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349–366.  
  9. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).  
  10. P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14 (2001) 263–278 (electronic).  
  11. P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations. Comm. Math. Phys. 216 (2001) 663–686.  
  12. M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry. Springer (2000).  
  13. J. Douglas, Jr., and T.F. Russell, 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 (1982) 871–885.  
  14. T.F. Dupont and Y. Liu, Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations. SIAM J. Numer. Anal. 40 (2002) 914–927 (electronic).  
  15. M. Falcone and R. Ferretti, Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35 (1998) 909–940 (electronic).  
  16. Y. Liu, R.E. Bank, T.F. Dupont, S. Garcia and R.F. Santos, Symmetric error estimates for moving mesh mixed methods for advection-diffusion equations. SIAM J. Numer. Anal. 40 (2003) 2270–2291.  
  17. I. Malcevic and O. Ghattas, Dynamic-mesh finite element method for Lagrangian computational fluid dynamics. Finite Elem. Anal. Des. 38 (2002) 965–982.  
  18. H. Masahiro, H. Katsumori and K. Mutsuto, Lagrangian finite element method for free surface Navier-Stokes flow using fractional step methods. Int. J. Numer. Methods Fluids13 (1991) 841–855.  
  19. K. Miller, Moving finite elements. II. SIAM J. Numer. Anal. 18 (1981) 1033–1057.  
  20. K. Miller and R.N. Miller, Moving finite elements. I. SIAM J. Numer. Anal. 18 (1981) 1019–1032.  
  21. K.W. Morton, A. Priestley and E. Süli, Stability of the Lagrange-Galerkin method with nonexact integration. RAIRO Modél. Math. Anal. Numér. 22 (1988) 625–653.  
  22. J. Ruppert, A new and simple algorithm for quality 2-dimensional mesh generation, in Third Annual ACM-SIAM Symposium on Discrete Algorithms (1992) 83–92.  
  23. V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics25. Springer-Verlag, Berlin (1997).  

Citations in EuDML Documents

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  1. Konstantinos Chrysafinos, Sotirios P. Filopoulos, Theodosios K. Papathanasiou, Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system
  2. Konstantinos Chrysafinos, Sotirios P. Filopoulos, Theodosios K. Papathanasiou, Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system
  3. Konstantinos Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

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