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, Volume 58, Issue 4, pp 317–334 | Cite as

A posteriori error control in radiative transfer

  • C. Führer
  • G. Kanschat
Article

Abstract

This note introduces a finite element approach for a radiative transfer model equation which allows for a posteriori error control and thus opens up the way for adaptive grid refinement strategies. Apart from a posteriori error estimates for the mean intensity, we demonstrate corresponding results for the intensity and compare our error estimators with a new, simple and rather promising error indicator for hyperbolic problems. We specifically emphasize that our ‘pure’ finite element technique is equivalent to the well-established discrete ordinates method favored by many users.

AMS Subject Classifications

45K05 65N15 65N30 65N50 

Key words

Radiative transfer neutron transport Boltzmann equation finite element methods a posteriori error estimates 

A Posteriori Fehlerkontrolle für Strahlungstransportprobleme

Zusammenfassung

In dieser Arbeit wird ein Finite Elemente Ansatz für eine Strahlungstransportmodellgleichung vorgestellt, der a posteriori Fehlerkontrolle gestattet und somit den Weg für adaptive Gittersteurung ebnet. Neben a posteriori Fehlerabschätzungen für die gemittelte Intensität wird eine entsprechende Abschätzung für die Einzelintensitäten abgeleitet und die so gewonnenen Fehlerschätzer schließlich mit einem neuen, recht einfachen und vielversprechenden Fehlerindikator für hyperbolische Probleme verglichen. Insbesondere ist die hier vorgestellte ‘reine’ Finite Elemente Technik im wesentlichen äquivalent zu der von vielen Anwendern benutzten diskreten Ordinaten Methode.

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References

  1. [1]
    Asadzadeh, M.: Convergence analysis of some numerical methods for neutron transport and Vlasov equations. Ph.D. thesis, Chalmers University of Technology, Göteborg, Schweden, 1986.Google Scholar
  2. [2]
    Auer, L.H.: Difference equations and linearization methods for radiative transfer methods in radiative transfer. In: Numerical radiative transfer (Kalkofen, W., ed.), pp. 101. Cambridge: Cambridge University Press 1984.Google Scholar
  3. [3]
    Eriksson, K., et al.: Adaptive finite element methods. Amsterdam: North-Holland 1996.Google Scholar
  4. [4]
    Eriksson, K., Johnson, C.: An adaptive finite element method for linear elliptic problems. Math. Comp.50, 361–383 1988.zbMATHCrossRefGoogle Scholar
  5. [5]
    Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal.28, 43–77 (1991)zbMATHCrossRefGoogle Scholar
  6. [6]
    Eriksson, J., Johnson, C.: Adaptive streamline diffusion finite element methods for convection-diffusion problems. Preprint 1990-18, Göteborg University, Sweden, 1990.Google Scholar
  7. [7]
    Führer, C.: Finite-Elemente-Diskretisierungen zur Lösung der 2D-Strahlungstrans-portgleichung, Diploma Thesis, Heidelberg University, 1993.Google Scholar
  8. [8]
    Führer, C., Rannacher, R.: Error analysis for the finite element approximation of a radiative transfer model. Preprint 94-46, SFB 359, Heidelberg University, 1994 (to appear in M2AN).Google Scholar
  9. [9]
    Johnson, C., Nävert, U.: Analysis for the finite element methods for advection-diffusion problems. Technical report Nr. 80.01, Chalmers University of Technology, Göteborg, Sweden.Google Scholar
  10. [10]
    Johnson, C., Pitkäranta, J.: Convergence of a fully discrete scheme for two-dimensional neutron transport. SIAM J. Numer. Anal.20, 951–966 (1983).zbMATHCrossRefGoogle Scholar
  11. [11]
    Kanschat, G.: Parallele und Adaptive Algorithmen für Strahlungstransportprobleme. Ph.D. Thesis, Heidelberg University, 1995.Google Scholar
  12. [12]
    Kanschat, G.: Parallel algorithms for radiative transfer problems. Conference Proceedings, Projects in Massively Parallel Computing 1994, Tech. Report 009-94, Paderborn Center for Parallel Computing.Google Scholar
  13. [13]
    Papkalla, R.: Linienentstehung in Akkretionsscheiben. Ph.D. Thesis, Heidelberg University, 1993.Google Scholar
  14. [14]
    Pitkäranta, J.: On the differential properties of solutions to Fredholm equations with weakly singular kernels. J. Inst. Math. Appl.24, 109–119 (1979).zbMATHCrossRefGoogle Scholar
  15. [15]
    Turek, S.: A generalized mean intensity approach for the numerical solution of the radiative transfer equation. Preprint 94-04, SFB 359, Heidelberg University, 1994.Google Scholar
  16. [16]
    Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comp. (to appear).Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • C. Führer
    • 1
  • G. Kanschat
    • 1
  1. 1.Sonderforschungsbereich 359Universität Heidelberg, INF 294HeidelbergGermany

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