The concept


We focus on a long-term program of collaborative efforts aimed at reducing the complexity of such systems by
revealing, exploiting, and preserving mathematical structures
. The term structure is used here to indicate some relevant background framing of the PDE system such as conservation or dissipation of physical quantities, gradient-flow evolution, asymptotic behavior, qualitative properties of trajectories, emergence of scale effects.

The
tenet of this SFB
is that taking full advantage of such structures plays a pivotal role for the successful treatment of complex PDE systems. This theme
cuts
across modeling, analysis, numerical analysis, and implementation and necessitates collaboration across traditional field boundaries. For example, in one direction qualitative properties of solutions, conservation of physical quantities, correct asymptotic behavior at the continuum level must be effectively reproduced by approximations in order to achieve improved performances. In the other direction, the need of block-based simulations calls for the development of efficient solvers, which in turn are strongly affected by problem formulations and ultimately by modeling choices.

Such an extensive, intradisciplinary scientific effort is only possible with the appropriate organizational structure. The SFB features a
totally integrated
scientific program based on a network of complementary expertises. This is realized by
Joint Work Packages (JWPs)
connecting two or more research groups. The JWP network interlaces the whole SFB and bridges over the whole span of competencies required by the project (from modeling to analysis and stochastics, to numerics and software development). In particular, it strictly connects the more analytically-oriented groups to the more numerically- and computationally-oriented, bringing to fruition the diversity of the SFB team expertises.



The project


The efficient modeling, analysis, and simulation of phenomena described by systems of partial differential equations (PDE systems) regularly calls for taming their inherent mathematical complexity. Examples of such complex PDE systems of relevance to us are quantum and electronic systems, thermomechanics and electromagnetism of solids and structures, biological systems and transportation networks. The interplay of different physical effects (multiphysics), the occurrence of distinguished time- and length-scales (multiscale), the competition of multiple components, the presence of diverse geometrical settings are often at the origin of mathematical criticalities making the study of these systems extremely demanding.

This SFB aims at a methodologically integrated development of the analytical, numerical, and computational treatment of complex PDE systems.

We propose to focus on a combination of research themes under the common vision of reducing the complexity of such systems by revealing, exploiting, and preserving mathematical structures. We use the term structure to indicate some relevant background framing of the PDE system such as conservation or dissipation of physical quantities, gradient-flow evolution, asymptotic behavior, qualitative properties of trajectories, emergence of scale effects. The tenet of this SFB is that the understanding of such structures plays a pivotal role for the successful treatment of the PDE system. These are crucial to connect accurate modeling and analysis to efficient approximations and reliable simulations. We unfold this global strategy by targeting three main research focuses:

  • - Variational and entropic structures in model reduction and asymptotics;
  • - Structure-preserving and problem-adapted approximations;
  • - Block-based natural-language simulations in multiphysics.

These themes are cooperatively tackled within eleven Project Parts, the core organizational concept being that of creating a new, totally interlaced research network.



More info about our the projects of the group can be found at this link and on the individual pages of the members of the SFB.