Authors: | Bäcker, Jan-Phillip Röger, Matthias Kuzmin, Dmitri |
Title: | Analysis and numerical treatment of bulk-surface reaction-diffusion models of Gierer-Meinhardt type |
Language (ISO): | en |
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. |
Subject Headings: | reaction-diffusion systems flux-corrected transport positivity preservation finite element method pattern formation PDEs on surfaces |
Subject Headings (RSWK): | Finite Elemente |
URI: | http://hdl.handle.net/2003/39810 http://dx.doi.org/10.17877/DE290R-21701 |
Issue Date: | 2020-10 |
Appears in Collections: | Ergebnisberichte des Instituts für Angewandte Mathematik |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
Ergebnisbericht Nr. 633.pdf | DNB | 3.21 MB | Adobe PDF | View/Open |
This item is protected by original copyright |
This item is protected by original copyright rightsstatements.org