%0 Book Section %1 citeulike:13979008 %A Rannacher, R. %B Error Control and Adaptivity in Scientific Computing %D 1999 %E Bulgak, Haydar %E Zenger, Christoph %I Springer %K 93c40-adaptive-control 65n30-pdes-bvps-finite-elements 65n15-pdes-bvps-error-bounds 68t05-learning-and-adaptive-systems %P 247--278 %R 10.1007/978-94-011-4647-0\_12 %T Error Control in Finite Element Computations %U http://dx.doi.org/10.1007/978-94-011-4647-0\_12 %V 536 %X We present a general paradigm for a posteriori error control and adaptive mesh design in finite element Galerkin methods. The conventional strategy for controlling the error in finite 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 finite element spaces and the stability properties of a linearized dual problem. The mesh refinement 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 coefficients. This deficiency 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 first for a simple model case and then illustrated by results for more complex problems in fluid mechanics, elasto-plasticity and radiative transfer. %@ 978-0-7923-5809-1

@incollection{citeulike:13979008, abstract = {{We present a general paradigm for a posteriori error control and adaptive mesh design in finite element Galerkin methods. The conventional strategy for controlling the error in finite 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 finite element spaces and the stability properties of a linearized dual problem. The mesh refinement 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 coefficients. This deficiency 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 first for a simple model case and then illustrated by results for more complex problems in fluid mechanics, elasto-plasticity and radiative transfer.}}, added-at = {2017-06-29T07:13:07.000+0200}, author = {Rannacher, R.}, biburl = {https://www.bibsonomy.org/bibtex/2947f7ad61825f839c0e790a7d203f0c4/gdmcbain}, booktitle = {Error Control and Adaptivity in Scientific Computing}, citeulike-article-id = {13979008}, citeulike-attachment-1 = {rannacher_99_error.pdf; /pdf/user/gdmcbain/article/13979008/1058545/rannacher_99_error.pdf; e72cf6e72e63d86f183e11856f528564d2dea9af}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/978-94-011-4647-0\_12}, citeulike-linkout-1 = {http://link.springer.com/chapter/10.1007/978-94-011-4647-0\_12}, comment = {(private-note)Cited by 'DROPS':file:///usr/local/src/sources/drops/doc/dox/html/bib.html\#B4Rannacher2}, doi = {10.1007/978-94-011-4647-0\_12}, editor = {Bulgak, Haydar and Zenger, Christoph}, file = {rannacher_99_error.pdf}, interhash = {7a2647ed582aad8ae5f85c85540d3532}, intrahash = {947f7ad61825f839c0e790a7d203f0c4}, isbn = {978-0-7923-5809-1}, issn = {1389-2185}, keywords = {93c40-adaptive-control 65n30-pdes-bvps-finite-elements 65n15-pdes-bvps-error-bounds 68t05-learning-and-adaptive-systems}, pages = {247--278}, posted-at = {2016-03-13 23:24:18}, priority = {2}, publisher = {Springer}, series = {NATO Science Series, Series C: Mathematical and Physical Sciences}, timestamp = {2019-05-01T06:15:32.000+0200}, title = {{Error Control in Finite Element Computations}}, url = {http://dx.doi.org/10.1007/978-94-011-4647-0\_12}, volume = 536, year = 1999 }