## Section: Overall Objectives

### Objectives

The main objective of this project is the development of innovative algorithms and efficient software tools for the simulation of complex flow problems. Accurate predictions of physical quantities are of great interest in fluid mechanics, for example in order to analyze instabilities, 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.

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. Our contributions concern modern discretization methods (high-order and adaptivity) and goal-oriented simulation tools (prediction of physical quantities, numerical sensitivities, and parameter identification). Concrete applications originate from aerodynamics, viscoelastic flows, heat transfer, and porous media.

The goal of the **first phase** of the project is to develop
flow solvers based on modern numerical methods such as high-order
discretization in space and time and self-adaptive
algorithms. Adaptivity based on a posteriori error estimators has
become a new paradigm in scientific computing, first because of
the necessity to give rigorous error bounds, and second because of
the possible speed-up of simulation tools. A systematic approach
to these questions requires an appropriate variational framework
and the development of a posteriori error estimates and adaptive
algorithms, as well as sufficiently general software tools able to
realize these algorithms. To this end we develop a single common
library written in C++ and study at hand of concrete applications
the possible benefits and difficulties related to these algorithms
in the context of fluid mechanics. The main ingredients of our
numerical approach are adaptive finite element discretizations
combined with multilevel solvers and hierarchical modeling. We
develop different kinds of finite element methods, such as
discontinuous (DGFEM) and stabilized finite element methods
(SFEM), either based on continuous or non-conforming finite
element spaces (NCFEM). The availability of such tools is also a
prerequisite for testing advanced physical models, concerning for
example turbulence, compressibility effects, and realistic models
for viscoelastic flows.

The goal of the **second phase** is to tackle questions going
beyond forward numerical simulations: parameter identification,
design optimization, and questions related to the interaction
between numerical simulations and physical experiments. It
appears that many questions in the field of complex flow problems
can neither be solved by experiments nor by simulations alone. In
order to improve the experiment, the software has to be able to
provide information beyond the results of simple simulation.
Here, information on sensitivities with respect to selected
measurements and parameters is required. The parameters could in
practice be as different in nature as a diffusion coefficient and
a velocity boundary condition. It is our long-term objective to
develop the necessary computational framework and to contribute to
the rational interaction between simulation and experiment.

The interdisciplinary collaboration is at the heart of this project. The team consists of mathematicians and physicists, and we develop collaborations with computer scientists.