Contemporary Mathematics Adaptive Finite Element Methods in Flow Computations

This article surveys recent developments of theory-based methods for mesh adaptivity and error control in the numerical solution of flow problems. The emphasis is on viscous incompressible flows governed by the Navier-Stokes equations. But also inviscid transsonic flows and viscous lowMach number flows including chemical reactions are considered. The Galerkin finite element method provides the basis for a common rigorous a posteriori error analysis. A large part of the existing work on a posteriori error analysis deals with error estimation in global norms such as the ‘energy norm’ involving usually unknown stability constants. However, in most CFD applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Such ‘goal-oriented’ error bounds can be derived by duality arguments borrowed from optimal control theory. These 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 ’dualweighted-residual method’ (DWR method), is developed within an abstract functional analytic setting, thus providing the general guideline for applications to various kinds of flow models including also aspects of flow control and hydrodynamic stability. Several examples are discussed in order to illustrate the main features of the DWR method.