A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples

A reened approach to residual-based error control in nite element (FE) discretiza-tions is presented. The conventional strategies for adaptive mesh reenement in FE methods are mostly based on a posteriori error estimates in the global energy or L 2-norm involving local residuals of the computed solution. The mesh reenement process then aims at equilibrating these local error indicators. Such estimates reeect the approximation properties of the nite element space by local interpolation constants while the stability properties of the continuous model enter through a global stability constant, which may be known explicitly in simple cases. Meshes generated on the basis of such global error estimates may not be appropriate in cases of strongly varying coeecients and for the computation of local quantities as, for example, point values or contour integrals. More detailed information about the mechanism of error propagation can be obtained by employing duality arguments specially adapted to the quantity of interest. This results in a posteriori error estimates in which the local information derived from the dual solution is used in the form of weights multiplied by local residuals. On the basis of such estimates, a feedback process in which the weights are numerically computed with increasing accuracy leads to almost optimal meshes for various kinds of error functionals. This approach is developed here for a simple model problem, namely the Poisson equation in two dimensions, in order to emphasize its basic features. However, the underlying concept is rather universal and has, on a heuristic basis, already been successfully applied to much more complex problems in structural and uid mechanics as well as in astrophysics.