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International Conference on Patterns of Dynamics

PaDy 2016: Patterns of Dynamics pp 242–268Cite as

Numerical Center Manifold Methods

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 205))

Abstract

This paper summarizes the first available proof and results for general full, so space and time discretizations for center manifolds of nonlinear parabolic problems. They have to admit a local time dependent solution (a germ) near the bifurcation point. For the linearization (A, B) of the nonlinear elliptic part and the boundary condition we require: The spectrum of A is located \(\ge - \beta \) for a small \(\beta > 0\) instead of the usual = 0, A is elliptic and for (A, B) the complementing condition is valid, hence A is sectorial. Indeed the two last conditions hold by Amann’s [3] criteria and remark for \(A:W^{m,p} (\Omega ,\mathbb {R}^q) \rightarrow W^{-m,p}, 1\le m,q,\) satisfying the Legendre-Hadamard condition and in appropriate divergence form for \(m>1.\) This does not apply to the generalized Agmon e.al. systems. By the active research, the class of problems satisfying the above conditions is strongly growing. Then, with geometric time discretizations, essentially all the up-to-date numerical space, except meshfree methods, yield converging numerical results for these “approximate” center manifolds. Here I summarize results of my upcoming monograph and strongly generalize my earlier papers.

Dedicated to the 60\(\mathrm{{th}}\) birthday of my good friend Prof. Dr. Bernold Fiedler.

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Notes

  1. 1.

    The following results are, with minor changes, e.g., the curve C in (13), correct for the open [53] and closed [32] sectors: \(\Sigma _{c, \vartheta } :=\{ \vartheta < |\arg (\lambda -c)| \cdots \}\) and \(\Sigma _{c, \vartheta } :=\{ \vartheta \le |\arg (\lambda -c)| \cdots \}.\)

  2. 2.

    Thanks to Robert Denk I found this paper strongly related to my problems.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Universität Marburg, Hans Meerwein Strasse, Lahnberge, 35032, Marburg, Germany

    Klaus Böhmer

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  1. Klaus Böhmer
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Editors and Affiliations

  1. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Pavel Gurevich

  2. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Juliette Hell

  3. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    Björn Sandstede

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Arnd Scheel

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Böhmer, K. (2017). Numerical Center Manifold Methods. In: Gurevich, P., Hell, J., Sandstede, B., Scheel, A. (eds) Patterns of Dynamics. PaDy 2016. Springer Proceedings in Mathematics & Statistics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-64173-7_15

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