Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-28T13:35:15.511Z Has data issue: false hasContentIssue false

An optimal control approach to a posteriori error estimation in finite element methods

Published online by Cambridge University Press:  09 January 2003

Roland Becker
Affiliation:
Institut für Angewandte Mathematik, Universität Heidelberg, INF 293/294, D-69120 Heidelberg, Germany http://gaia.iwr.uni-heidelberg.de E-mail: Roland.Becker@iwr.uni-heidelberg.de, Rolf.Rannacher@iwr.uni-heidelberg.de
Rolf Rannacher
Affiliation:
Institut für Angewandte Mathematik, Universität Heidelberg, INF 293/294, D-69120 Heidelberg, Germany http://gaia.iwr.uni-heidelberg.de E-mail: Roland.Becker@iwr.uni-heidelberg.de, Rolf.Rannacher@iwr.uni-heidelberg.de

Abstract

This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.

Type
Research Article
Copyright
© Cambridge University Press 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)