zbMATH — the first resource for mathematics

A posteriori error control in radiative transfer. (English) Zbl 0880.65125
The authors obtain an aposteriori error estimate for the mean density of a radiative transfer model equation by a suitable finite element approach. The method is based on a duality argument that guarantees reliable error control. Numerical tests are considered to confirm the indicator’s reliability and efficiency on regular quadrilateral grids. It is claimed that the finite element technique used here is equivalent to the well-established discrete ordinates method.

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
85A25 Radiative transfer in astronomy and astrophysics
Full Text: DOI
[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. · Zbl 0608.65098
[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.
[3] Eriksson, K., et al.: Adaptive finite element methods. Amsterdam: North-Holland 1996.
[4] Eriksson, K., Johnson, C.: An adaptive finite element method for linear elliptic problems. Math. Comp.50, 361–383 1988. · Zbl 0644.65080 · doi:10.1090/S0025-5718-1988-0929542-X
[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) · Zbl 0732.65093 · doi:10.1137/0728003
[6] Eriksson, J., Johnson, C.: Adaptive streamline diffusion finite element methods for convection-diffusion problems. Preprint 1990-18, Göteborg University, Sweden, 1990. · Zbl 0795.65074
[7] Führer, C.: Finite-Elemente-Diskretisierungen zur Lösung der 2D-Strahlungstrans-portgleichung, Diploma Thesis, Heidelberg University, 1993.
[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). · Zbl 0866.65093
[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.
[10] Johnson, C., Pitkäranta, J.: Convergence of a fully discrete scheme for two-dimensional neutron transport. SIAM J. Numer. Anal.20, 951–966 (1983). · Zbl 0538.65097 · doi:10.1137/0720065
[11] Kanschat, G.: Parallele und Adaptive Algorithmen für Strahlungstransportprobleme. Ph.D. Thesis, Heidelberg University, 1995.
[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.
[13] Papkalla, R.: Linienentstehung in Akkretionsscheiben. Ph.D. Thesis, Heidelberg University, 1993.
[14] Pitkäranta, J.: On the differential properties of solutions to Fredholm equations with weakly singular kernels. J. Inst. Math. Appl.24, 109–119 (1979). · Zbl 0423.45004 · doi:10.1093/imamat/24.2.109
[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. · Zbl 0822.65129
[16] Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comp. (to appear). · Zbl 0854.65094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.