Taming Complexity in Partial Differential Systems

A wide variety of phenomena from the Natural Sciences and Technology can be described in terms of systems of partial differential equations (PDEs). These systems often show a relevant level of inherent mathematical complexity: the interplay of different physical effects, the occurrence of distinguished time- and length-scales, the competition of multiple components, the presence of diverse geometrical settings often make the treatment of such systems very demanding. This SFB aims at breaking this complexity via an integrated development of the analytical, numerical, and computational treatment. 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 main methodological idea is that of revealing, exploiting, and preserving mathematical structures. The term `structure` is used to indicate some relevant background framing of the PDE system such as conservation or dissipation of physical quantities, specific evolution modes, asymptotic behavior with respect to time, qualitative properties of trajectories, emergence of scale effects, among others. 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 integration strategy by developing eleven Project Parts. The core organizational concept being that of creating a new, totally interlaced research network.