Error Control in Finite Element

We present a general paradigm for a posteriori error control and adap-tive mesh design in nite element Galerkin methods. The conventional strategy for controlling the error in nite element methods is based on a posteriori estimates for the error in the global energy or L 2-norm involving local residuals of the computed solution. Such estimates contain constants describing the local approximation properties of the nite element spaces and the stability properties of a linearized dual problem. The mesh reenement then aims at the equilibration of the local error indicators. However, meshes generated via controlling the error in a global norm may not be appropriate for local error quantities like point values or line integrals and in case of strongly varying coeecients. This deeciency may be overcome by introducing certain weight-factors in the a posteriori error estimates which depend on the dual solution and contain information about the relevant error propagation. This way, optimally economical meshes may be generated for various kinds of error measures. This is systematically developed rst for a simple model case and then illustrated by results for more complex problems in uid mechanics, elasto-plasticity and radiative transfer. Recommended literature and references: The basics on the mathematical theory of nite element methods used in this paper can be found in the books of Johnson 14] and Brenner, Scott 10]. An introduction into the general concept of residual-based error control for nite element methods has been given in the survey article by Eriksson, Estep, Hansbo, Johnson 11], and with some modiications in the papers by Becker, Rannacher 6], 7]. Surveys of the traditional approach to a posteriori error estimation are given by Verf urth 23] and Ainsworth, Oden 1]. The material of this paper has mainly been collected from the papers of Becker, For results of computations for special applications, we refer to the PhD theses of Becker 3] (incompressible ows), Kanschat 16] (radiative transfer problems), Suttmeier 22] (elasto-plasticity problems), and Braack 9], Waguet 24] ((ows with chemical reactions).