Lagrangian and moving mesh methods for the convection diffusion equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, p. 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 : https://doi.org/10.1051/m2an:2007053
Classification:  65M60,  65M15
Keywords: convection diffusion, moving meshes, lagrangian formulation
@article{M2AN_2008__42_1_25_0,
     author = {Chrysafinos, Konstantinos and Walkington, Noel J.},
     title = {Lagrangian and moving mesh methods for the convection diffusion equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     pages = {25-55},
     doi = {10.1051/m2an:2007053},
     zbl = {1136.65089},
     mrnumber = {2387421},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_1_25_0}
}
Chrysafinos, Konstantinos; Walkington, Noel J. Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, pp. 25-55. doi : 10.1051/m2an:2007053. http://www.numdam.org/item/M2AN_2008__42_1_25_0/

[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 0622.76031

[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 1202654 | Zbl 0770.65060

[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 1897945 | Zbl 1014.65087

[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 1862992 | Zbl 0989.65122

[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 1616666 | Zbl 0911.65088

[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 1862993 | Zbl 0989.65123

[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 2206444 | Zbl 1110.65088

[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 2217386 | Zbl 1112.65086

[9] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058

[10] P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14 (2001) 263-278 (electronic). | MR 1815212 | Zbl 0997.76009

[11] P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations. Comm. Math. Phys. 216 (2001) 663-686. | MR 1815721 | Zbl 0988.76020

[12] M. De Berg, M. Van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry. Springer (2000). | MR 1763734 | Zbl 0939.68134

[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 672564 | Zbl 0492.65051

[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 1949398 | Zbl 1025.65051

[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 1619910 | Zbl 0914.65097

[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 1974185 | Zbl 1038.65085

[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 1906245 | Zbl 1059.76036

[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 0741.76037

[19] K. Miller, Moving finite elements. II. SIAM J. Numer. Anal. 18 (1981) 1033-1057. | MR 638997 | Zbl 0518.65083

[20] K. Miller and R.N. Miller, Moving finite elements. I. SIAM J. Numer. Anal. 18 (1981) 1019-1032. | MR 638996 | Zbl 0518.65082

[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 974291 | Zbl 0661.65114

[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 1213221 | Zbl 0801.68163

[23] V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics 25. Springer-Verlag, Berlin (1997). | MR 1479170 | Zbl 0528.65052