Adaptive finite element method for unsteady convection-dominated flows with stiff source terms

SIAM Journal of Scientific Computing

Volumn 21, Issue 3, 1999, Pages 799-818

  Hebeker, Friedrich K ^a   Rannacher, Rolf ^b  

^a JUSTUS LIEBIG UNIVERSITY   (Germany)

^b UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; COMPUTATIONAL FLUID DYNAMICS; CONFORMAL MAPPING; ERROR ANALYSIS; GALERKIN METHODS; OPTIMIZATION; PROBLEM SOLVING; VECTORS;

CONVECTION DOMINATED FLOWS; DETONATION WAVES; STREAMLINE DIFFUSION;

FINITE ELEMENT METHOD;

EID: 0033294025     PISSN: 10648275     EISSN: None     Source Type: None    
DOI: 10.1137/S1064827597319039     Document Type: Article

References (33)

1. J. BEBERNES AND D. EBERLY, Mathematical Problems from Combustion Theory, Springer, New York, 1989.
2. R. BECKER AND R. RANNACHER, Weighted a posteriori error control in FE methods, in Proceedings ENUMATH-97, Heidelberg, 1997, World Scientific, Singapore, 1998.
3. R. BECKER AND R. RANNACHER, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math., 4 (1996), pp. 237-264.
4. K. BÖTTCHER AND R. RANNACHER, Adaptive Error Control in Solving Ordinary Differential Equations by the Discontinuous Galerkin Method, Preprint 96-53, SFB 359, Universität Heidelberg, Heidelberg, Germany, 1996.
5. A. J. BOURLIOUX, A. J. MAJDA, AND V. ROYTBURD, Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J. Appl. Math., 51 (1991), pp. 303-343.
6. P. COLELLA, A. MAJDA, AND V. ROYTBURD, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Statist. Comp., 7 (1986), pp. 1059-1080.
7. R. COURANT AND K. O. FRIEDRICHS, Supersonic Flow and Shock Waves, Interscience, New York, 1948.
8. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43-77.
9. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems. IV. Nonlinear Problems, SIAM J. Numer. Anal., 32 (1995), pp. 1729-1749.
10. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Introduction to adaptive methods for differential equations, Acta Numerica, 1995, pp. 105-159.
11. C. FÜHRER AND R. RANNACHER, An adaptive streamline-diffusion finite element method for hyperbolic conservation laws, East-West J. Numer. Math., 5 (1997), pp. 145-162.
12. R. GATHMANN AND F. K. HEBEKER, Numerical problems of a convection diffusion equation with Arrhenius source term, in Proceeding of the Fifth European Conference on Mathematics in Industry, M. Heiliö, ed., Teubner, Stuttgart, 1991, pp. 229-233.
13. E. GODLEWSKI AND P. A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, Berlin, Heidelberg, New York, 1996.
14. F. K. HEBEKER, An Adaptive Finite Element Method for Detonation, Waves: Towards Reaction Time Limit Zero, Preprint 98-27, SFB 359, Universität Heidelberg, Heidelberg, Germany. 1998.
15. F. K. HEBEKER, R. MALY, AND S. SCHOFFEL, Numerical simulation of knock damage phenomena, on an IBM RS/6000 workstation cluster, in Proceedings 26th International Symposium on Automotive Technology and Automation, Automotive Automation, Croydon, 1993, pp. 275-282.
16. J. HILDTICH AND P. COLELLA, A front tracking method for compressible flames in one dimension, SIAM J. Sci. Comput., 16 (1995), pp. 755-772.
17. C. HIRSCH, Numerical Computations of Internal and External Flows, Wiley, Chichester, 1990.
18. T. J. R. HUGHES, L. P. FRANCA, AND M. MALLET, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg., 54 (1986), pp. 223-234.
19. C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, UK, 1987.
20. C. JOHNSON AND A. SZEPESSY, Adaptive finite element methods for conservation laws based on a posteriori error estimates, Comm. Pure Appl. Math., 48 (1995). pp. 199-234.
21. G. KANSCHAT, Solution of multi-dimensional radiative transfer problems on parallel computers, in Parallel Solution of Partial Differential Equations, IMA Vol. Math. Appl., Springer-Verlag, New York, to appear.
22. R. LEVEQUE, Numerical Methods for Conservations Laws, 2nd ed., Birkhäuser Verlag, Basel, 1990.
23. R. J. LEVEQUE AND K. M. SHUYE, One-dimensional front tracking based on high resolution wave propagation methods, SIAM J. Sci. Comput., 16 (1995), pp. 348-377.
24. R. LEVEQUE AND H. YEE, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys., 86 (1990), pp. 187-210.
25. J. D. MURRAY, On the Gunn effect and other physical examples of perturbed conservation laws, J. Fluid Mech., 44 (1970), pp. 315-346.
26. R. B. PEMBER, Numerical methods for hyperbolic conservation laws with stiff relaxation. II. Higher-order Godunov methods, SIAM J. Sci. Comput., 14 (1993), pp. 824-859.
27. R. RANNACHER, A posteriori error estimation in least-squares stabilized finite element schemes, Comput. Methods Appl. Mech. Engrg., 166 (1998), pp. 99-114.
28. R. RANNACHER AND F.-T. SUTTMEIER, A feed-back approach to error control in finite element methods: Application to linear elasticity, Comput. Mech., 19 (1997), pp. 434-446.
29. R. RANNACHER AND G. ZHOU, Pointwise superconvergence of the streamline diffusion finite-element method, Numer. Methods Partial Differential Equations, 12 (1996), pp. 123-145.
30. R. SANDBOGE, Adaptive Finite Element Methods for Reactive Flow Problems, Ph.D. thesis, Chalmers University of Technology, Gothenburg, 1996.
31. K. STREHMEL AND R. WEINER, Numerik gewöhnlicher Differentialgleichungen, Teubner, Stuttgart, 1995.
32. W. A. YONG, Singular Perturbations of First-Order Hyperbolic Systems, Ph.D. thesis, SFB 359, Universität Heidelberg, Heidelberg, Germany, 1992.
33. G. ZHOU, How accurate is the streamline diffusion finite element method?, Math. Comp., 66 (1996), pp. 31-44.