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An adaptive multi-level method for convection diffusion problems. (English) Zbl 0952.65067
From the authors’ abstract: 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.

MSC:
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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References:
[1] M. Bercovier, O. Pironneau and V. Sastri, Finite elements and characteristics for some parabolic-hyperbolic problems. Appl. Math. Modelling7 (1983) 89-96. · Zbl 0505.65055 · doi:10.1016/0307-904X(83)90118-X
[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).
[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. · Zbl 0897.76070 · doi:10.1137/S0036142994267989
[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). · Zbl 0941.65101
[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. · Zbl 0838.76060 · doi:10.1007/BF00312446
[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 · Zbl 0492.65051 · doi:10.1137/0719063
[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. · Zbl 0732.65093 · doi:10.1137/0728003
[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 · Zbl 0663.76054 · eudml:193526
[10] P. Houston and E. Suli, Adaptative Lagrange-Galerkin methods for unsteady convection-dominated diffusion problems, Oxford University Computing Laboratory Report, 95/24 (1995).
[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 · Zbl 0953.65065 · doi:10.1002/(SICI)1098-2426(200001)16:1<107::AID-NUM8>3.0.CO;2-0
[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. · Zbl 0853.65092 · doi:10.1137/0732054
[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). · Zbl 0665.73059
[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 · Zbl 0637.76024 · doi:10.1007/BF01396329 · eudml:133286
[17] E. Suli and A.F. Ware, A spectral method of characteristics for hyperbolic problems. SIAM. J. Numer. Anal.28 (1991) 423-445. · Zbl 0743.65080 · doi:10.1137/0728024
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