含内质网神经元钙动力学模型的存在唯一性及隐-显有限元格式
Modeling calcium dynamics in neurons with endoplasmic reticulum: existence, uniqueness and an implicit-explicit finite element scheme
摘要：
像许多其他生物过程一样，含有内质网的神经元中的钙动力学受界面分离域上的扩散反应方程控制。交界条件通常由提供界面通量的常微分方程系统描述。以钙模型为例，利用比较定理、抛物算子的基本解和Picard存在性定理中的一个技巧，证明了这类常微分方程-通量边界界面问题解的存在性、唯一性和有界性.然后我们提出并分析了一种高效的隐-显有限元格式，该格式对抛物算子是隐式的，对非线性项是显式的。我们表明，稳定性不依赖于空间的网格步长。并给出了在$H^1$范数下的最优收敛速度。数值实验验证了理论结果。
英文摘要：
Like many other biological processes, calcium dynamics in neurons containing an endoplasmic reticulum 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 the calcium model as an example of this class of ODE-flux boundary interface problems, we prove the existence, uniqueness and boundedness of the solution 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.
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