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A posteriori error estimates for the monodomain model in cardiac electrophysiology

Luca Ratti, Marco Verani

TL;DR

This work develops a posteriori error estimators for the monodomain cardiac electrophysiology model, formulated as a parabolic equation for the transmembrane potential $u$ coupled with a nonlinear ODE for a recovery variable $w$, and discretized with a Newton-Galerkin scheme. By decomposing the total residual into linearization, time discretization, and space discretization components, the authors prove reliability and, under a coercivity-type assumption on the nonlinearities, efficiency of the estimators in the $L^2(0,T;H^1())$-type norm. They also derive a priori estimates in the $L^2(0,T;H^1())$ framework and validate the theory via numerical experiments showing accurate error upper bounds and linear convergence with respect to both mesh size and time step. The results enable robust space-time adaptive strategies that can accelerate simulations and aid inverse problems such as ischemic region identification in cardiac tissue.

Abstract

We consider the monodomain model, a system of a parabolic semilinear reaction-diffusion equation coupled with a nonlinear ordinary differential equation, arising from the (simplified) mathematical description of the electrical activity of the heart. We derive a posteriori error estimators accounting for different sources of error (space/time discretization and linearization). We prove reliability and efficiency (this latter under a suitable assumption) of the error indicators. Finally, numerical experiments assess the validity of the theoretical results.

A posteriori error estimates for the monodomain model in cardiac electrophysiology

TL;DR

This work develops a posteriori error estimators for the monodomain cardiac electrophysiology model, formulated as a parabolic equation for the transmembrane potential coupled with a nonlinear ODE for a recovery variable , and discretized with a Newton-Galerkin scheme. By decomposing the total residual into linearization, time discretization, and space discretization components, the authors prove reliability and, under a coercivity-type assumption on the nonlinearities, efficiency of the estimators in the -type norm. They also derive a priori estimates in the framework and validate the theory via numerical experiments showing accurate error upper bounds and linear convergence with respect to both mesh size and time step. The results enable robust space-time adaptive strategies that can accelerate simulations and aid inverse problems such as ischemic region identification in cardiac tissue.

Abstract

We consider the monodomain model, a system of a parabolic semilinear reaction-diffusion equation coupled with a nonlinear ordinary differential equation, arising from the (simplified) mathematical description of the electrical activity of the heart. We derive a posteriori error estimators accounting for different sources of error (space/time discretization and linearization). We prove reliability and efficiency (this latter under a suitable assumption) of the error indicators. Finally, numerical experiments assess the validity of the theoretical results.

Paper Structure

This paper contains 10 sections, 7 theorems, 108 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. A Newton-Galerkin scheme for the approximation of the monodomain model
  3. A priori estimates for the space semidiscretization
  4. Residual operators
  5. A posteriori estimators
  6. Efficiency of the estimators
  7. Numerical experiments
  8. Spatial and temporal analysis
  9. Linearization analysis
  10. Conclusions

Key Result

Proposition 2.1

Let the initial data $u_0 \in C^{2+\alpha}(\bar{\Omega})$, $w_0 \in C^{\alpha}(\bar{\Omega})$ satisfy the bound $0 \leq u_0 \leq 1$ and $0 \leq w_0 \leq \frac{A(1+a)^2}{4}$, consider $M \in C^2(\Omega)$ and let the following compatibility conditions hold: $M \nabla u_0 \cdot \nu = 0$, being $\partia

Figures (4)

  • Figure 1: Snapshots of the evolution of the electrical potential. In Figures (a)-(f) the contour plots are shown in some selected instants $t_1, \ldots, t_6$. Figure (g) reports the value of the electrical potential in a specific point $P$; the instants $t_1, \ldots, t_6$ are remarked.
  • Figure 2: Assessment of the upper bound
  • Figure 3: Convergence analysis in $h$ and $\tau$
  • Figure 4: Assessment of the a posteriori indicator $\gamma_k^n$ for the linearization error

Theorems & Definitions (17)

  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • ...and 7 more