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