论文标题
Gierer-Meinhardt类型的散装表面反应扩散模型的分析和数值处理
Analysis and numerical treatment of bulk-surface reaction-diffusion models of Gierer-Meinhardt type
论文作者
论文摘要
我们在表面上考虑了一个吉列 - 梅恩哈特系统，并在散装中与抛物线PDE结合，该域被该表面限制。最近提出了这种模型，并通过Gomez，Ward和Wei对二维散装域进行了分析（Siam J.Appl。Dyn。Syst。18，2019年）。我们证明了在任意空间尺寸中散装系统的良好性，并表明溶液一直在抛物线Hölder空间中均匀界定。证明使用Schauders固定点定理，并在表面和散装部分中分裂。我们还求解了一个还原的系统，对应于数值混合的整体解决方案的假设。我们使用操作员分解方法，结合了Laplace-Beltrami操作员的有限元离散化以及对源和下沉术语的阳性治疗方法。所提出的方法基于通量校正的转运（FCT）范式。它以提供阳性保存，质量保护和平滑区域的二阶精度的方式约束减少问题的空间和时间离散化。该系统在二维球体上的数值研究结果表明，在一维模型中也观察到了局部稳态多型模式。
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 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.
