FOR 3013: Vector- and Tensor-Valued Surface PDEs

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PDEs on surfaces remain an active field of research in applied mathematics. Due to their coupling with the geometry such surface PDEs are intrinsically nonlinear. This leads to new challenges in modeling and numerical analysis. Most of these challenges are addressed for scalar-valued surface PDEs. In the scalar case the coupling between surface geometry and the PDE is relatively weak, and thus numerical approaches established in flat spaces are applicable after small modifications. For vector- and tensor-valued surface PDEs these approaches are no longer sufficient. The surface vector- and tensor-fields often need to fulfill additional constraints. One example is the tangentiality of these fields, for which they need to be considered as elements of the tangent bundle, leading to a nonlinear coupling between the surface geometry and the PDE. To deal with these new challenges was the motivation for this research unit. Within the last three years we have seen a tremendous growth in research activities, with development of new models, new numerical methods, new numerical analysis results, new software tools, and new applications. Our research unit has significantly contributed to these developments and in several fields also initiated them. In the second funding period we will deepen this knowledge for a description and understanding that uncovers universal principles and makes similarities and differences between the considered applications transparent. Different from the first funding period a focus will be on surface dynamics, including deformable fluid surfaces, rate-independent evolution of prestrained plates, growth and swelling phenomena and the development of numerical methods and their analysis for these new challenges. Addressing these challenges can only be explored by combining various mathematical disciplines for which this research unit is perfectly suited. We strengthen our expertise in numerical analysis and mechanics and based on the success of the first funding period expect our results to further foster this fast growing research field within mathematics and other disciplines to enable breakthrough developments.