First, we describe some typical difficulties in our fields of application which require the improvement of established and the development of new methods. Then we give the research directions underlying our project for the development of new software tools. They are summarized under the three headlines 'High-order methods', 'Adaptivity', and 'Parameter identification and numerical sensitivities'.

Accurate predictions of physical quantities are of great interest in fluid mechanics, for example in order to analyze instabilities in reacting flows, predict forces acting on a body, estimate the flow through an orifice, or predict thermal conductivity coefficients. Due to the complex and highly nonlinear equations to be solved, it is difficult to know in advance how fine the spatial or temporal resolution should be and how detailed a given physical model has to be represented. We propose to develop a systematic approach to these questions based on auto-adaptive methods.

We note that most of the physical problems under consideration have a three-dimensional character and involve the coupling of models and extremely varying scales. This makes the development of fast numerical methods and efficient implementation a question of feasibility.

The modeling of reactive flows , and turbulent reactive flows , implies a number of difficulties.

In addition to the classical challenges of DNS in very simple situations, the extension to more involved geometries and boundary conditions raises new questions in this field.

Recently, it has been observed that certain turbulence models have similarities to finite element stabilization techniques for the Navier-Stokes equations. Such variational multi-scale methods lead to adaptive turbulence modeling, which are however far from being a standard tool for turbulent flow simulations nowadays. We intend to contribute to this development, and the development of our DNS code is an important ingredient.

Despite numerous efforts, numerical simulation of polymer flows is still a very challenging research area. There exist only relatively few commercial codes for the simulation of these flows (PolyFlow ^®, Flow3D or Rem3D). Two reasons seem to be responsible for this :

A major issue to be addressed is the breakdown in convergence of the algorithms at critical values of the Weissenberg number. The commercial codes are only able to deal with Weissenberg numbers up to 10, which means that the behavior of the fluid is not very elastic. This limit is too low to describe the polymer flow in a processing machine and is often explained by difficulties related to the numerical schemes. In particular, it has been widely believed that the high Weissenberg number problem is attributed to the loss of the positivity of the conformation tensor C at the discrete level. Note that even if the positive-definiteness of C is always true at the continuous level, it is very difficult to extend it to the discrete counterpart since the conformation tensor is not a direct unknown of the approximated problem.

Recently, different new attempts heave been made to overcome these difficulties. Lee and Xu made an important step in the comprehension of how discretizations preserving the positive-definiteness of the conformation tensor C can be derived. The idea is to write the constitutive law in terms of the conformation tensor and to recast it (by using certain Lie derivatives) into the formulation of symmetric Riccati differential equations. It is then natural to use the well developed theory for the approximation of Riccati equations, which arise in many other fields of applied mathematics such as optimal control, differential geometry or singular perturbation theory. However, one drawback of their approach is that it relies on the use of the characteristic method for the treatment of the convection term, which, in turn, presents several known drawbacks. So far, it has not yet been shown that this approach leads to a robust method for practical problems.

Another approach, which has attracted much attention, is the introduction of the logarithm of the conformation tensor by Fattal & Kupferman . The main idea is, that the matrix-exponential will always yield a symmetric positive-definite matrix, even if a non-monotone scheme is used for its approximation. However, in order to do so, the constitutive law has first to be expressed in terms of the logarithm of the conformation tensor. Although preliminary computational studies indicate a gain in stability, it is not so clear, what the impact of this nonlinear transformation, which can be viewed as a scale compression, on the numerical approximation is.

Besides this fundamental difficulty related to the Weissenberg number, several other aspects are to be taken into account when simulating polymer flows: large number of unknowns (pressure, velocity and stress at least), nonlinear character, treatment of the convection terms (especially in the constitutive law), strong thermo-mechanical coupling, and three-dimensionality of flows. We note that most studies on numerical tools for viscoelastic flows deal with isothermal two-dimensional flow (planar or axisymmetric). It remains a major issue to find stable and robust numerical methods capable to deal with 3D anisothermal flows at Weissenberg numbers greater than 10, in the frame of realistic models.

The discontinuous Galerkin finite element method (DGFEM) offers interesting perspectives, since it gives a framework for the combination of techniques developed in the incompressible finite-element (well-founded treatment of incompressibility constraints, pressure approximation, and stabilization for high-Reynolds-number flows) and the compressible finite-volume community (entropy solutions, Riemann solvers and flux limiters).

In addition, the order limit of finite volume discretizations is broken by the variational formulation underlying DGFEM, which makes it possible to develop discretization schemes with local mesh refinement and local variation of the polynomial degree ( hp-methods). At the same time, the well-established finite-element knowledge for saddle-point problems can be set on work.

Noting that different approaches based on discontinuous Galerkin methods have been used in recent years for the solution of challenging flow problems. Since the project team members have experience with these and other stabilized finite element methods, a combination of the different techniques is expected to be beneficial in order to gain efficiency.

It is generally accepted that an important advantage of DGFEM beside its flexibility is the fact that it is locally conservative. At the same time, its drawback is its relatively high numerical cost. For example, compared to continuous P¹finite elements on a triangular mesh, the number of unknowns are increased by a factor of 6 (and a factor of 2 with respect to the Crouzeix-Raviart space); considering the system matrix even leads to a more disadvantageous count. Concerning higher-order spaces, standard DGFEM has a negligible overhead for polynomial orders starting from p= 5, which is probably not the most employed in practice. The question of how to increase efficiency of DGFEM is an important topic of recent research. Our approach in this field is based on comparison with stabilized FE methods.

There are many similarities between DGFEM and SDFEM (streamline-diffusion FEM) based on piecewise linear elements for the transport equation from a theoretical point of view, see for example the classical text book . Recently, attempts have been made to shed brighter light on the relations between these methods , . A better understanding of the relations between these methods will contribute to the development of more efficient schemes with desired properties. As outlined before, the goal is to cut down the computational overhead of standard DGFEM, while retaining its robustness and conservation properties.

Formulation of discretization schemes based on discontinuous finite element spaces is nowadays standard. However, some important questions remain to be solved:

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 , , based on two-sided a posteriori error estimators . The situation is completely open for p-adaptivity, model-adaptivity or the DWR method Dual-weighted-residual method , 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 ; see also the overview paper , application to laminar reacting flows in , and application to the Euler equations in . 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 . Recently, it has been extended to the control of modeling errors . 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.

Numerical simulations generally involve parameters of different nature. Some parameters reflect physical properties of the materials under consideration, or describe the way they interact. In addition to these parameters the values of which are often determined by experiments and sometimes only known with accuracy under certain conditions, the development of a computational model involves additional quantities, which could for example be related to boundary and initial conditions.

The generalization of the DWR method to parameter identification problems has been developed in , and for time-dependent equations. The case of finite-dimensional parameters, which is theoretically less challenging than the infinite-dimensional case and has therefore been less treated in the literature, is of particular interest in view of the presented applications (for example the estimation of a set of diffusion velocities).

The goal of numerical simulations are in general the computation of given output values Iwhich are obtained from the approximated physical fields by additional computations, often termed post-processing. The DWR method places these output values in the center of interest and aims at providing reliable and efficient computations of these quantities.

In the context of calibration of parameter values with experiments, it seems to be natural to go one step beyond the sole computation of I. Indeed, the computation of numerical sensitivities or condition numbers I/ q_iwhere q_idenotes a single parameter can be expected to be of practical and theoretical interest, either in order to improve the design of experiments, or in order to help to analyze the outcome of an experiment.

It turns out, that similar techniques as those employed for parameter identification can be used in order to obtain information on parameter sensitivities and corresponding a posteriori error analysis .