Goal-Oriented A Posteriori Error Estimation for Multiple Target Functionals

R. Hartmann and P. Houston

Abstract:

In many applications the quantities of interest are a series of target functionals of the solution to the governing system of partial differential equations rather than the solution itself. For example, in the field of aerodynamics, examples include the drag and lift coefficients of an airfoil immersed into a fluid, the pressure difference between the leading and trailing edges of the airfoil and point evaluations of the density or pressure on the profile of the airfoil. While traditionally these quantities are measured in wind tunnel experiments, nowadays these experiments are increasingly replaced by numerical simulations aiming to predict these quantities to a high level of accuracy.
In a series of previous articles, we have developed the theory of goal--oriented a posteriori error estimation for discontinuous Galerkin methods applied to inviscid compressible fluid flows. On the basis of Type I a posteriori bounds we considered the design of adaptive finite element algorithms that are capable of generating optimal meshes specifically tailored to the efficient computation of a single target functional of practical interest. The purpose of the current article is to extend this earlier work to the case when several target functionals of the solution need to be simultaneously approximated to a given level of accuracy.