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dc.contributor.authorBäcker, Jan-Phillip-
dc.contributor.authorRöger, Matthias-
dc.contributor.authorKuzmin, Dmitri-
dc.description.abstractWe consider a Gierer-Meinhardt system on a surface coupled with aparabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (SIAM J. Appl. Dyn. Syst. 18, 2019).We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The proof uses Schauders fixed point theorem and a splitting in a surface and a bulk part. We also solve a reduced system, corresponding to the assumption of a well mixed bulk solution, numerically. We use operator-splitting methods which combine a finite element discretization of the Laplace-Beltrami operator with a positivity-preserving treatment of the source and sink terms. The proposed methodology is based on the flux-corrected transport (FCT) paradigm. It constrains the space and time discretization of the reduced problem in a manner which provides positivity preservation, conservation of mass, and second-order accuracy in smooth regions. The results of numerical studies for the system on a two-dimensional sphere demonstrate the occurrence of localized steady-state multispike pattern that have also been observed in one-dimensional models.en
dc.relation.ispartofseriesErgebnisberichte des Instituts für Angewandte Mathematik;633-
dc.subjectreaction-diffusion systemsen
dc.subjectflux-corrected transporten
dc.subjectpositivity preservationen
dc.subjectfinite element methoden
dc.subjectpattern formationen
dc.subjectPDEs on surfacesen
dc.titleAnalysis and numerical treatment of bulk-surface reaction-diffusion models of Gierer-Meinhardt typeen
dc.subject.rswkFinite Elementede
dcterms.accessRightsopen access-
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