Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems

SIAM Journal on Numerical Analysis

Volumn 39, Issue 6, 2002, Pages 1954-1984

  Bause, Markus ^a   Knabner, Peter ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

uniform convergence; Characteristics; Convection diffusion equation; Finite element method; Lagrange Galerkin scheme

Indexed keywords

EID: 0036556440     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142900367478     Document Type: Article

References (35)

1. M. BAUSE, Optimum Convergence Rates for Fully Discretized Navier-Stokes Approximations of Higher Order in Lipschitz Domains, Ph.D. thesis, University Paderborn, Paderborn, Germany, 1997 (in German).
2. M. BAUSE AND P. KNABNER, Uniform Error Analysis for Lagrange-Galerkin Approximations of Convection-Dominated Diffusion Problems. Part I: A Priori Analysis in Lagrangian Coordinates and Optimal Order Error Estimates for the Time Discretization, preprint 256, Inst. Appl. Math., University Erlangen-Nuremberg, Erlangen, Germany, 1999.
3. M. BAUSE AND P. KNABNER, On Uniform convergence rates for Eulerian and Lagrangian finite element approximations of convection-dominated diffusion problems, Comput. Methods Appl. Math., submitted.
4. M. BAUSE, Scales of ε-uniform a priori estimates for nonstationary convection-dominated diffusion problems, Ann. Univ. Ferrara Sez. VII Sc. Mat., 46 (2000), pp. 115-137.
5. M. BAUSE AND P. KNABNER, Uniform Error Analysis for Lagrange-Galerkin Approximations of Convection-Dominated Diffusion Problems. Part II: Time Discretization for Rough Initial-Values, Error Estimates for the Spatial Discretization and Numerical Results, preprint 270, Inst. Appl. Math., University Erlangen-Nuremberg, Erlangen, Germany, 2000.
6. M. BAUSE, On optimum convergence rates for higher order Navier-Stokes approximations. Part I: Error estimates for the spatial discretization, Numer. Math., submitted.
7. P. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
8. A. J. CHORIN AND J. E. MARSDEN, A Mathematical Introduction to Fluid Mechanics, Springer, New York, 1984.
9. 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), pp. 871-885.
10. J. DOUGLAS, C. S. HUANG, AND F. PEREIRA, The modified method of characteristics with adjusted advection, Numer. Math., 83 (1999), pp. 353-369.
11. R. E. EWING, ED., The Mathematics of Reservoir Simulation, SIAM, Philadelphia, 1983.
12. H. FUJITA AND T. KATO, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), pp. 269-315.
13. A. FRIEDMAN, Partial Differential Equations, Krieger, New York, 1976.
14. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
15. P. HARTMAN, Ordinary Differential Equations, Wiley, New York, 1973.
16. J. G. HEYWOOD AND R. RANNACHER, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), pp. 275-311.
17. J. G. HEYWOOD AND R. RANNACHER, Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), pp. 353-384.
18. C., HIRSCH, Numerical Computation of Internal and External Flows, Vol. 1 Wiley, Chichester 1988.
19. M. JAKOB, Heat Transfer, Wiley, New York, 1959.
20. C. JOHNSON, R. RANNACHER, AND M. BOMAN, Numerics and hydrodynamic stability: Toward error control in computational fluid dynamics, SIAM J. Numer, Anal., 32 (1995), pp. 1058-1079.
21. T. KATO, Perturbation Theory for Linear Operators, Springer, Berlin, 1976.
22. P. KNABNER, I. KÖGEL-KNABNER, AND K. U, TOTSCHE, The modeling of reactive solute transport with sorption to mobile and immobile sorbents. Part 1: Experimental evidence and model development, Water Resources Research, 32 (1996), pp. 1611-1622.
23. S. G. KREIN, J. I. PETUNIN, AND E. M. SEMENOV, Interpolation for Linear Operators, Transl. Math. Monogr. 54, AMS, Providence, RI, 1982.
24. G. E. LADAS AND V. LAKSHMIKANTHAM, Differential Equations in Abstract Spaces, Academic Press, New York, 1972.
25. J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Problems and Applications, Vol. I, Springer, Berlin, 1972.
26. K. W. MORTON, A. PRIESTLEY, AND E. SÜLI, Stability of the Lagrange-Galerkin method with non-exact integration, RAIRO Modél Math. Anal. Numér., 22 (1988), pp. 625-653.
27. K. W. MORTON AND E. SÜLI, Evolution-G'alerkin methods and their supraconvergence, Numer. Math., 71 (1995), pp. 331-355.
28. O. PIRONNEAU, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38 (1982), pp. 309-332.
29. 2-convergent approximation schemes to the Navier-Stokes equations, Comment. Math. Univ. St. Paul., 43 (1994). pp. 55-108.
30. H. G. ROOS, M. STYNES, AND L. TOBISKA, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996.
31. E. SÜLI, Stability and convergence of the Lagrange-Galerkin method with non-exact integration, in The Mathematics of Finite Elements and Applications VI, J. R. Whiteman, ed., Academic Press, London, 1988, pp. 459-483.
32. H. TANABE, Equations of Evolution, Pitman, London, 1979.
33. H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
34. H. WANG, R. E. EWING, AND T. F. RUSSELL, Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis, IMA J. Numer. Anal., 15 (1995), pp. 405-459.
35. J. WLOKA, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.