Analysis and numerical treatment of bulk-surface reaction-diffusion models of Gierer-Meinhardt type
| dc.contributor.author | Bäcker, Jan-Phillip | |
| dc.contributor.author | Röger, Matthias | |
| dc.contributor.author | Kuzmin, Dmitri | |
| dc.date.accessioned | 2020-11-09T13:13:42Z | |
| dc.date.available | 2020-11-09T13:13:42Z | |
| dc.date.issued | 2020-10 | |
| dc.description.abstract | We 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.identifier.issn | 2190-1767 | |
| dc.identifier.uri | http://hdl.handle.net/2003/39810 | |
| dc.identifier.uri | http://dx.doi.org/10.17877/DE290R-21701 | |
| dc.language.iso | en | |
| dc.relation.ispartofseries | Ergebnisberichte des Instituts für Angewandte Mathematik;633 | |
| dc.subject | reaction-diffusion systems | en |
| dc.subject | flux-corrected transport | en |
| dc.subject | positivity preservation | en |
| dc.subject | finite element method | en |
| dc.subject | pattern formation | en |
| dc.subject | PDEs on surfaces | en |
| dc.subject.ddc | 610 | |
| dc.subject.rswk | Finite Elemente | de |
| dc.title | Analysis and numerical treatment of bulk-surface reaction-diffusion models of Gierer-Meinhardt type | en |
| dc.type | Text | |
| dc.type.publicationtype | preprint | |
| dcterms.accessRights | open access | |
| eldorado.secondarypublication | false |