Lagrangian and moving mesh methods for the convection diffusion equation
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 25-55.

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.

DOI : 10.1051/m2an:2007053
Classification : 65M60, 65M15
Mots-clés : convection diffusion, moving meshes, lagrangian formulation 
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     author = {Chrysafinos, Konstantinos and Walkington, Noel J.},
     title = {Lagrangian and moving mesh methods for the convection diffusion equation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {25--55},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
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     mrnumber = {2387421},
     zbl = {1136.65089},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2007053/}
}
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Chrysafinos, Konstantinos; Walkington, Noel J. Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 25-55. doi : 10.1051/m2an:2007053. http://archive.numdam.org/articles/10.1051/m2an:2007053/

[1] R. Balasubramaniam and K. Mutsuto, Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Numer. Methods Fluids 7 (1987) 953-984. | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[4] J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L 2 projection in H 1 (Ω). Math. Comp. 71 (2002) 147-156 (electronic). | MR | Zbl

[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). | MR | Zbl

[6] C. Carstensen, Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H 1 -stability of the L 2 -projection onto finite element spaces. Math. Comp. 71 (2002) 157-163 (electronic). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[9] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR | Zbl

[10] P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14 (2001) 263-278 (electronic). | MR | Zbl

[11] P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations. Comm. Math. Phys. 216 (2001) 663-686. | MR | Zbl

[12] M. De Berg, M. Van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry. Springer (2000). | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[17] I. Malcevic and O. Ghattas, Dynamic-mesh finite element method for Lagrangian computational fluid dynamics. Finite Elem. Anal. Des. 38 (2002) 965-982. | MR | Zbl

[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 Fluids 13 (1991) 841-855. | Zbl

[19] K. Miller, Moving finite elements. II. SIAM J. Numer. Anal. 18 (1981) 1033-1057. | MR | Zbl

[20] K. Miller and R.N. Miller, Moving finite elements. I. SIAM J. Numer. Anal. 18 (1981) 1019-1032. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[23] V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics 25. Springer-Verlag, Berlin (1997). | MR | Zbl

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