## Section: Scientific Foundations

### Numerical tools: Adaptivity

The possible benefits of local mesh refinement for fluid dynamical problems is nowadays uncontested; the obvious arguments are the presence of singularities, shocks, and combustion fronts. The use of variable polynomial approximation is more controversial in CFD, since the literature does not deliver a clear answer concerning its efficiency. At least at view of some model problems, the potential gain obtained by the flexibility to locally adapt the order of approximation is evident. It remains to investigate if this estimation stays true for the applications to be considered in the project.

The design and analysis of auto-adaptive methods as described above is a recent research topic, and only very limited theoretical results are known. Concerning the convergence of adaptive methods for mesh refinement, only recently there has been made significant progress in the context of the Poisson problem [45] , [60] [5] , based on two-sided a posteriori error estimators [66] . The situation is completely open for p -adaptivity, model-adaptivity or the DWR method(Dual-weighted-residual method [2] , see below.). In addition, not much seems to be known for nonlinear equations. It can be hoped that theoretical insight will contribute to the development of better adaptive algorithms. We note that most of the mathematical literature deals with a posteriori error estimators in the energy norm related to linear symmetric problems.

A more praxis-oriented approach to error estimation is the DWR method, developed in [3] ; see also the overview paper [2] , application to laminar reacting flows in [40] , and application to the Euler equations in [55] . The idea of the DWR method is to consider a given, user-defined physical quantity as a functional acting on the solution space. This allows the derivation of a posteriori error estimates which directly control the error in the approximation of the functional value. This approach has been applied to local mesh-refinement for a wide range of model problems [2] . Recently, it has been extended to the control of modeling errors [47] . The estimator of the DWR method requires the computation of an auxiliary linear partial differential equation. So far, relatively few research has been done in order to use possibly incomplete information from, e.g., coarse discretization of this equation.