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Adaptive finite element approximation of hyperbolic problems. (English) Zbl 1141.76428
Barth, Timothy J. (ed.) et al., Error estimation and adaptive discretization methods in computational fluid dynamics. Berlin: Springer (ISBN 3-540-43758-4/hbk). Lect. Notes Comput. Sci. Eng. 25, 269-344 (2003).
Summary: We review some recent developments concerning the a posteriori error analysis of $$h$$- and $$hp$$-version finite element approximations to hyperbolic problems. The error bounds stem from an error representation formula which equates the error in an output functional of interest to the inner product of the finite element residual with the solution of a dual (adjoint) problem whose data is the density function of the target functional. Type I a posteriori error bounds are derived which, unlike the cruder Type II bounds, retain the dual solution in the bound as a local weight-function. The relevance of Type I a posteriori bounds is argued by showing that the local size of the error in a hyperbolic problem may be only very weakly correlated to the local size of the residual; consequently, adaptive refinement algorithms based on the size of the local residual alone can be ineffective. The sharpness of Type I a posteriori error bounds is demonstrated on both structured and adaptively refined meshes.
For the entire collection see [Zbl 0999.00014].

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76N15 Gas dynamics, general