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A posteriori error estimates and adaptive methods for hyperbolic and convection dominated parabolic conservation laws. (English) Zbl 1076.35072

Kirkilionis, Markus (ed.) et al., Trends in nonlinear analysis. On the occasion of the 60th birthday of Willi Jäger. Berlin: Springer (ISBN 3-540-44198-0/hbk). 289-306 (2003).
This paper gives a survey of recent theoretical results on a posteriori error estimates and adaptive mesh refinement for finite volume methods applied to hyperbolic conservation laws and convection dominated parabolic conservation laws. The authors consider scalar hyperbolic problems, weakly coupled hyperbolic conservation laws such as transport of contaminant in porous media. The cell-centered finite volume methods as well as staggered Lax-Friedrichs scheme are used for hyperbolic problems, whereas the dual finite volume methods using linear approximations on primary triangular mesh for diffusion trems is applied for the convection dominated weakly coupled systems. The results of a posteriori error estimates are presented for these three models. Numerical experiments presented at the end of the paper for transport of contaminants with degradation confirm the theoretical results.
For the entire collection see [Zbl 1001.00073].

MSC:

35L65 Hyperbolic conservation laws
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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