An Adaptive Multi-level method for Convection Diffusion Problems

Martine Marion; Adeline Mollard

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 2, page 439-458
  • ISSN: 0764-583X

Abstract

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In this article we introduce an adaptive multi-level method in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of different time-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a corresponding adaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments, in particular for a convection-dominated problem.

How to cite

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Marion, Martine, and Mollard, Adeline. " An Adaptive Multi-level method for Convection Diffusion Problems." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 439-458. <http://eudml.org/doc/197405>.

@article{Marion2010,
abstract = { In this article we introduce an adaptive multi-level method in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of different time-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a corresponding adaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments, in particular for a convection-dominated problem. },
author = {Marion, Martine, Mollard, Adeline},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Multi-level methods; convection-diffusion equations; characteristics method; a posteriori error estimates; adaptive algorithms.; adaptive multi-level method; convection diffusion problems; characteristics method; error estimate; numerical experiments},
language = {eng},
month = {3},
number = {2},
pages = {439-458},
publisher = {EDP Sciences},
title = { An Adaptive Multi-level method for Convection Diffusion Problems},
url = {http://eudml.org/doc/197405},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Marion, Martine
AU - Mollard, Adeline
TI - An Adaptive Multi-level method for Convection Diffusion Problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 2
SP - 439
EP - 458
AB - In this article we introduce an adaptive multi-level method in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of different time-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a corresponding adaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments, in particular for a convection-dominated problem.
LA - eng
KW - Multi-level methods; convection-diffusion equations; characteristics method; a posteriori error estimates; adaptive algorithms.; adaptive multi-level method; convection diffusion problems; characteristics method; error estimate; numerical experiments
UR - http://eudml.org/doc/197405
ER -

References

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  1. M. Bercovier, O. Pironneau and V. Sastri, Finite elements and characteristics for some parabolic-hyperbolic problems. Appl. Math. Modelling7 (1983) 89-96.  Zbl0505.65055
  2. K. Boukir, Y. Maday, B. Metivet and R. Razafindrakoto, A high-order characteristics/finite element method for imcompressible Navier-Stokes equations, Rapport de l'Université Pierre et Marie Curie, R 92032 (1992).  Zbl0904.76040
  3. J.B. Burie and M. Marion, Multi-level methods in space and time for Navier-Stokes equations. SIAM J. Numer. Anal.34 (1997) 1574-1599.  Zbl0897.76070
  4. J.B. Burie and M. Marion, Adaptative multi-level methods in space and time for paraboloc problems- The periodic case. Math. of Comp. (to appear).  Zbl0941.65101
  5. A. Debussche, T. Dubois and R. Temam, The nonlinear Galerkin method: A multi-scale method applied to the simulation of turbulent flows. Theoret. Comput. Fluid Dynamics7 (1995) 279-315.  Zbl0838.76060
  6. J. Douglas and T.F. Russel, Numerical methods for convection dominated diffusion problems based on combining the method of caracteristics with finite element methods or finite difference method. SIAM J. Numer. Anal.19 (1982) 871-885.  Zbl0492.65051
  7. T. Dubois, Simulation numérique d'écoulement homogènes et non-homogènes par des méthodes multi-résolution, Thèse, Université Paris-Sud (1993).  
  8. K. Eriksson and C. Johnson, Adaptative finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal.28 (1991) 43-77.  Zbl0732.65093
  9. C. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. M2AN22 (1998) 93-114.  Zbl0663.76054
  10. P. Houston and E. Suli, Adaptative Lagrange-Galerkin methods for unsteady convection-dominated diffusion problems, Oxford University Computing Laboratory Report, 95/24 (1995).  Zbl0957.65085
  11. F. Jauberteau, Résolution numérique des équations de Navier-Stokes instationnaires par méthodes spectrales. Méthode de Galerkin non linéaire, Thèse, Université Paris-Sud (1990).  
  12. M. Marion and A. Mollard, A multi-level characteristics method for periodic convection-dominated diffusion problems. Numer. Meth. PDEs. (to appear).  Zbl0953.65065
  13. M. Marion and J. Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal.32 (1995) 1170-1184.  Zbl0853.65092
  14. A. Mollard, Méthodes de caractéristiques multi-niveaux en espace et en temps pour une équation de convection-diffusion - Cas d'une approximation pseudo-spectrale, Thèse, École Centrale de Lyon (1998).  
  15. O. Pironneau, Finite element methods for fluids, Masson (1989).  Zbl0665.73059
  16. E. Suli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes Equations. Numer. Math.53 (1988) 459-483.  Zbl0637.76024
  17. E. Suli and A.F. Ware, A spectral method of characteristics for hyperbolic problems. SIAM. J. Numer. Anal.28 (1991) 423-445.  Zbl0743.65080

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