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

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.