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
%0 Generic
%1 bansch1995posteriori
%A Bänsch, Eberhard
%A Siebert, Kunibert G.
%D 1995
%K imported vorlaeufig
%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
@misc{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 = {2018-07-20T10:54:15.000+0200},
author = {B{\"a}nsch, Eberhard and Siebert, Kunibert G.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/262f11e92038fc325d54945c0985baa94/mhartmann},
howpublished = {Freiburg, Preprint 30/1995},
interhash = {d19f49d45b6e6f3a7d0736d24a44aa46},
intrahash = {62f11e92038fc325d54945c0985baa94},
keywords = {imported vorlaeufig},
owner = {kohlsk},
timestamp = {2018-07-20T08:54:15.000+0200},
title = {A Posteriori Error Estimation for Nonlinear Problems by Duality Techniques},
year = 1995
}