| Issue |
ESAIM: M2AN
Volume 57, Number 4, July-August 2023
|
|
|---|---|---|
| Page(s) | 2595 - 2621 | |
| DOI | https://doi.org/10.1051/m2an/2023052 | |
| Published online | 08 August 2023 | |
Numerical analysis of finite element methods for the cardiac extracellular-membrane-intracellular model: Steklov–Poincaré operator and spatial error estimates
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada
* Corresponding author: ybourg@uottawa.ca
Received:
13
July
2022
Accepted:
5
June
2023
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.
Mathematics Subject Classification: 65M12 / 65M15 / 65M60 / 35K57 / 92C37
Key words: xtracellular-membrane-intracellular model / Numerical methods / Manufactured solution / Steklov–Poincaré operator / Semi-implicit backward differentiation formula / Cardiac electrophysiology
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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