Substructuring Methods in Nonlinear Function Spaces

  • Oliver SanderEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


We generalize substructuring methods to problems for functions v: Ω → M, where Ω is a domain in \(\mathbb{R}^{d}\) and M is a Riemannian manifold. Examples for such functions include configurations of liquid crystals, ferromagnets, and deformations of Cosserat materials. We show that a substructuring theory can be developed for such problems. While the theory looks very similar to the linear theory on a formal level, the objects it deals with are much more general. In particular, iterates of the algorithms are elements of nonlinear Sobolev spaces, and test functions are replaced by sections in certain vector bundles. We derive various solution algorithms based on preconditioned Richardson iterations for a nonlinear Steklov–Poincaré formulation. Preconditioners appear as bundle homomorphisms. As a numerical example we compute the deformation of a geometrically exact Cosserat shell with a Neumann–Neumann algorithm.


Vector Bundle Trace Theorem Domain Decomposition Algorithm Test Function Space Interface Space 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany

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