# Lagrangian and moving mesh methods for the convection diffusion equation

Konstantinos Chrysafinos; Noel J. Walkington

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 1, page 25-55
- ISSN: 0764-583X

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topChrysafinos, Konstantinos, and Walkington, Noel J.. "Lagrangian and moving mesh methods for the convection diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 25-55. <http://eudml.org/doc/250339>.

@article{Chrysafinos2008,

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.
},

author = {Chrysafinos, Konstantinos, Walkington, Noel J.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Convection diffusion; moving meshes; Lagrangian formulation.; Lagrangian formulation; convection-diffusion equation; error estimates; finite element; convection dominated flows},

language = {eng},

month = {1},

number = {1},

pages = {25-55},

publisher = {EDP Sciences},

title = {Lagrangian and moving mesh methods for the convection diffusion equation},

url = {http://eudml.org/doc/250339},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Chrysafinos, Konstantinos

AU - Walkington, Noel J.

TI - Lagrangian and moving mesh methods for the convection diffusion equation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/1//

PB - EDP Sciences

VL - 42

IS - 1

SP - 25

EP - 55

AB -
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.

LA - eng

KW - Convection diffusion; moving meshes; Lagrangian formulation.; Lagrangian formulation; convection-diffusion equation; error estimates; finite element; convection dominated flows

UR - http://eudml.org/doc/250339

ER -

## References

top- R. Balasubramaniam and K. Mutsuto, Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Numer. Methods Fluids7 (1987) 953–984. Zbl0622.76031
- R.E. Bank and R.F. Santos, Analysis of some moving space-time finite element methods. SIAM J. Numer. Anal. 30 (1993) 1–18. Zbl0770.65060
- 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). Zbl1014.65087
- 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). Zbl0989.65122
- 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). Zbl0911.65088
- 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). Zbl0989.65123
- 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. Zbl1110.65088
- K. Chrysafinos and J.N. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349–366. Zbl1112.65086
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). Zbl0383.65058
- P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14 (2001) 263–278 (electronic). Zbl0997.76009
- P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations. Comm. Math. Phys. 216 (2001) 663–686. Zbl0988.76020
- M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational Geometry. Springer (2000). Zbl0939.68134
- 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. Zbl0492.65051
- 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). Zbl1025.65051
- 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). Zbl0914.65097
- 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. Zbl1038.65085
- I. Malcevic and O. Ghattas, Dynamic-mesh finite element method for Lagrangian computational fluid dynamics. Finite Elem. Anal. Des. 38 (2002) 965–982. Zbl1059.76036
- 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 Fluids13 (1991) 841–855. Zbl0741.76037
- K. Miller, Moving finite elements. II. SIAM J. Numer. Anal. 18 (1981) 1033–1057. Zbl0518.65083
- K. Miller and R.N. Miller, Moving finite elements. I. SIAM J. Numer. Anal. 18 (1981) 1019–1032. Zbl0518.65082
- 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. Zbl0661.65114
- 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. Zbl0801.68163
- V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics25. Springer-Verlag, Berlin (1997). Zbl0884.65097

## Citations in EuDML Documents

top- Konstantinos Chrysafinos, Sotirios P. Filopoulos, Theodosios K. Papathanasiou, Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system
- Konstantinos Chrysafinos, Sotirios P. Filopoulos, Theodosios K. Papathanasiou, Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system
- Konstantinos Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

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