A General Concept for Adaptivity in Finite Element Methods

Monday, April 28, 1997 - 11:15am - 12:00pm
Keller 3-180
Rolf Rannacher (Ruprecht-Karls-Universität Heidelberg)
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 L2-norm involving local residuals of the computed solution. Such estimates contain constants describing the local approximation properties of the finite-element space and the stability properties of a linearized dual problem. The mesh refinement then aims at equilibrating the local error indicators. However, meshes generated in this way via controlling the error in a global norm may not be appropriate for local error quantities like point values or contour integrals, and in the case of strongly varying coefficients. This deficiency may be overcome by introducing weight factors in the a posteriori error estimates which depend on the dual solution and contain information about the relevant error propagation. In this way optimal economical meshes can be obtained for various kinds of error measures. This will be illustrated for simple model cases as well as for a nonlinear problem in perfect plasticity. Another application is the computation of local quantities in viscous flows as, for example, drag and lift of a blunt body, which will be discussed in the talk by Roland Becker.