Numerical analysis of finite element methods for the cardiac extracellular-membrane-intracellular model: Steklov–Poincaré operator and spatial error estimates

IF 1.9 3区 数学 Q2 Mathematics
Diane Fokoué, Y. Bourgault
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引用次数: 0

Abstract

The extracellular-membrane-intracellular (EMI) model consists in a set of Poisson equations in two adjacent domains, coupled on interfaces with nonlinear transmission conditions involving a system of ODEs. The unusual coupling of PDEs and ODEs on the boundary makes the EMI models challenging to solve numerically. In this paper, we reformulate the problem on the interface using a Steklov–Poincaré operator. We then discretize the model in space using a finite element method (FEM). We prove the existence of a semi-discrete solution using a reformulation as an ODE system on the interface. We derive stability and error estimates for the FEM. Finally, we propose a manufactured solution and use it to perform numerical tests. The order of convergence of the numerical method agrees with what is expected on the basis of the theoretical analysis of the convergence.
心脏细胞外-膜-细胞内模型的有限元方法数值分析:steklov - poincar算子和空间误差估计
细胞外-膜-细胞内(EMI)模型由两个相邻区域的泊松方程组成,这些泊松方程耦合在具有非线性传输条件的界面上,涉及一个ode系统。边界上偏微分方程和偏微分方程的不寻常耦合使得电磁干扰模型的数值求解具有挑战性。在本文中,我们利用steklov - poincarcarr算子在界面上重新表述了这个问题。然后,我们使用有限元方法(FEM)在空间中离散模型。我们证明了半离散解的存在性,用一个重表述作为接口上的ODE系统。给出了有限元法的稳定性和误差估计。最后,我们提出了一个制造的解决方案,并使用它进行了数值测试。数值方法的收敛阶与理论分析的收敛阶一致。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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