Analysis and numerical treatment of bulk-surface reaction-diffusion models of Gierer-Meinhardt type

TL;DR

This paper analyzes a bulk-surface Gierer-Meinhardt reaction-diffusion model, proves its well-posedness in arbitrary dimensions, and develops a numerical scheme that preserves positivity and mass, demonstrating spike patterns on a sphere.

Contribution

It extends the analysis of bulk-surface Gierer-Meinhardt systems to arbitrary dimensions and introduces a positivity-preserving numerical method based on operator-splitting and flux-corrected transport.

Findings

Solutions remain bounded in parabolic Hölder spaces for all times.

Numerical simulations show localized spike patterns on a sphere.

The method ensures positivity, mass conservation, and second-order accuracy.

Abstract

We consider a Gierer-Meinhardt system on a surface coupled with a parabolic 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 H\"older 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.