Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T09:56:01.416Z Has data issue: false hasContentIssue false
  • English
  • Français

An Adaptive Multi-level method for Convection DiffusionProblems

Published online by Cambridge University Press:  15 April 2002

Martine Marion  and
Adeline Mollard
Martine Marion
Affiliation:
UMR CNRS 5585, Département Mathématique-Informatique, École Centrale de Lyon, BP 163, 69131 Écully Cedex, France.
Adeline Mollard
Affiliation:
UMR CNRS 5640, Département Génie Mathématique, INSA, Complexe de Rangueil, 31077 Toulouse Cedex 4, France.
Get access

Abstract

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

Keywords

Multi-level methodsconvection-diffusion equationscharacteristics methoda posteriori error estimatesadaptive algorithms.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 439 - 458
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bercovier, M., Pironneau, O. and Sastri, V., Finite elements and characteristics for some parabolic-hyperbolic problems. Appl. Math. Modelling 7 (1983) 89-96. CrossRef
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).
Burie, J.B. and Marion, M., Multi-level methods in space and time for Navier-Stokes equations. SIAM J. Numer. Anal. 34 (1997) 1574-1599. CrossRef
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).
Debussche, A., Dubois, T. and Temam, R., The nonlinear Galerkin method: A multi-scale method applied to the simulation of turbulent flows. Theoret. Comput. Fluid Dynamics 7 (1995) 279-315. CrossRef
Douglas, J. and Russel, T.F., 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. CrossRef
T. Dubois, Simulation numérique d'écoulement homogènes et non-homogènes par des méthodes multi-résolution, Thèse, Université Paris-Sud (1993).
Eriksson, K. and Johnson, C., Adaptative finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43-77. CrossRef
C. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. M 2 AN 22 (1998) 93-114.
P. Houston and E. Suli, Adaptative Lagrange-Galerkin methods for unsteady convection-dominated diffusion problems, Oxford University Computing Laboratory Report, 95/24 (1995).
F. Jauberteau, Résolution numérique des équations de Navier-Stokes instationnaires par méthodes spectrales. Méthode de Galerkin non linéaire, Thèse, Université Paris-Sud (1990).
M. Marion and A. Mollard, A multi-level characteristics method for periodic convection-dominated diffusion problems. Numer. Meth. PDEs. (to appear).
Marion, M. and Error, J. Xu estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32 (1995) 1170-1184. CrossRef
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).
O. Pironneau, Finite element methods for fluids, Masson (1989).
Suli, E., Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes Equations. Numer. Math. 53 (1988) 459-483. CrossRef
Suli, E. and Ware, A.F., A spectral method of characteristics for hyperbolic problems. SIAM. J. Numer. Anal. 28 (1991) 423-445. CrossRef