%0 Book %1 bansch1995posteriori %A Bänsch, Eberhard %A Siebert, Kunibert G. %B Preprintserie des Mathematischen Instituts %C Freiburg %D 1995 %I Mathematics Department, University of Freiburg %K %N 95-30 %T A Posteriori Error Estimation for Nonlinear Problems by Duality Techniques %X We present an abstract framework for a posteriori error estimation in finite element methods for a quite general class of nonlinear problems F(u) = 0 in X* where X is a Hilbert space, X* its dual and F: X -> X*. The error between exact and discrete solution is estimated in a weaker norm than the corresponding energy norm. Using the Aubin-Nitsche-trick, the error is represented by a linear dual problem. Assuming that the solution of the dual problem is regular, i. e. the solution belongs to a subspace W of X with a stronger norm, the error is estimated by the weaker W*-norm of the residual from above and below. Since it is not possible to compute the W*-norm of the residual, we also present utilities for an estimation of this norm for second order problems. In second order problems we want to estimate the error in the L^2-norm. For a localization of the residual we have to construct suitable cut-off functions which have weak second derivatives. This reflects the regularity of the dual problem on the discrete level and yields an estimation of the error by the error estimator from above and below. Concrete error estimators for second order semi-linear and eigenvalue problems are presented AMS-Classification. 65N30, 65N50 Freiburg, Preprint Nr. 30/1995

@book{bansch1995posteriori, abstract = {We present an abstract framework for a posteriori error estimation in finite element methods for a quite general class of nonlinear problems F(u) = 0 in X* where X is a Hilbert space, X* its dual and F: X -> X*. The error between exact and discrete solution is estimated in a weaker norm than the corresponding energy norm. Using the Aubin-Nitsche-trick, the error is represented by a linear dual problem. Assuming that the solution of the dual problem is regular, i. e. the solution belongs to a subspace W of X with a stronger norm, the error is estimated by the weaker W*-norm of the residual from above and below. Since it is not possible to compute the W*-norm of the residual, we also present utilities for an estimation of this norm for second order problems. In second order problems we want to estimate the error in the L^2-norm. For a localization of the residual we have to construct suitable cut-off functions which have weak second derivatives. This reflects the regularity of the dual problem on the discrete level and yields an estimation of the error by the error estimator from above and below. Concrete error estimators for second order semi-linear and eigenvalue problems are presented AMS-Classification. 65N30, 65N50 Freiburg, Preprint Nr. 30/1995}, added-at = {2023-09-01T10:41:50.000+0200}, address = {Freiburg}, author = {Bänsch, Eberhard and Siebert, Kunibert G.}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/279071aa0eaa868d8ab9a13e7bff5725f/unibiblio-4}, interhash = {3fd6bafb527a01fbd105448902f1aac2}, intrahash = {79071aa0eaa868d8ab9a13e7bff5725f}, keywords = {}, language = {eng}, number = {95-30}, publisher = {Mathematics Department, University of Freiburg}, series = {Preprintserie des Mathematischen Instituts}, timestamp = {2023-09-01T08:41:50.000+0200}, title = {A Posteriori Error Estimation for Nonlinear Problems by Duality Techniques}, year = 1995 }