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Lagrangian and moving mesh methods for the convection diffusion equation

Published online by Cambridge University Press:  12 January 2008

Konstantinos Chrysafinos  and
Noel J. Walkington
Konstantinos Chrysafinos
Affiliation:
: Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece. chrysafinos@ins.uni-bonn.de Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213 USA. noelw@cmu.edu
Noel J. Walkington
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213 USA. noelw@cmu.edu
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Abstract

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.

Keywords

Convection diffusionmoving meshesLagrangian formulation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 25 - 55
Copyright
© EDP Sciences, SMAI, 2008

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References

Balasubramaniam, R. and Mutsuto, K., Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Numer. Methods Fluids 7 (1987) 953–984.
R.E. Bank and R.F. Santos, Analysis of some moving space-time finite element methods. SIAM J. Numer. Anal. 30 (1993) 1–18.
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).
J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L2 projection in H1 (Ω). Math. Comp. 71 (2002) 147–156 (electronic).
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).
C. Carstensen, Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H1 -stability of the L2 -projection onto finite element spaces. Math. Comp. 71 (2002) 157–163 (electronic).
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.
K. Chrysafinos and J.N. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349–366.
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).
P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14 (2001) 263–278 (electronic).
P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations. Comm. Math. Phys. 216 (2001) 663–686.
M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry. Springer (2000).
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.
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).
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).
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.
I. Malcevic and O. Ghattas, Dynamic-mesh finite element method for Lagrangian computational fluid dynamics. Finite Elem. Anal. Des. 38 (2002) 965–982.
Masahiro, H., Katsumori, H. and Mutsuto, K., Lagrangian finite element method for free surface Navier-Stokes flow using fractional step methods. Int. J. Numer. Methods Fluids 13 (1991) 841–855.
K. Miller, Moving finite elements. II. SIAM J. Numer. Anal. 18 (1981) 1033–1057.
K. Miller and R.N. Miller, Moving finite elements. I. SIAM J. Numer. Anal. 18 (1981) 1019–1032.
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.
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.
V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics 25. Springer-Verlag, Berlin (1997).