Many systems, e.g. biological dynamics, are governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. Using cellular calcium dynamics as an example of this class of ODE-flux boundary interface problems we prove the existence, uniqueness and boundedness of the solutions by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard's existence theorem. Then we propose and analyze an efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in $H^1$ norm is obtained. Numerical experiments illustrate the theoretical results.

翻译：许多系统,例如生物动态系统,都受界面分离域域的分散-反应方程式的制约。界面条件通常被普通的差别方程式系统描述,这些方程式在界面之间提供通量。使用细胞钙动力作为这一类ODE-Flive边界界面问题的例子,我们通过应用比较理论、抛物线操作员的基本解决办法和Picard的存在理论中使用的战略,证明了解决方案的存在、独特性和相互交错性。然后我们提出和分析一个高效的隐含的、隐含的有限要素方案,这个方案对抛物线操作员是隐含的,对非线性术语是明确的。我们表明,稳定性并不取决于空间网状大小。我们还获得了1美元标准的最佳趋同率。数字实验显示了理论结果。