Key result
An adaptive version of the Newton algorithm based on a posteriori error estimates led to computational savings while ensuring solution accuracy for cardiac monodomain model simulations.
Why the study?
Numerical resolution of the cardiac monodomain model simulating electrical activity within heart tissue is often challenging and computationally expensive.
The proposed adaptive Newton algorithm provides computational savings while maintaining accuracy for simulating cardiac electrical activity using the monodomain model.
Enables certified accuracy in cardiac monodomain simulations; extends a posteriori analysis but leaves open clinical translation.
In this work, we consider the monodomain problem to simulate the electrical activity within the heart tissue. From a mathematical standpoint, it is modeled by a nonlinear parabolic partial differential equation coupled with an ordinary differential equation, whose numerical resolution is often challenging and computationally expensive. We employ the backward Euler scheme for the time discretization and the conforming Pp finite element method for the spatial discretization. We derive a guaranteed a posteriori error estimate on the error between the exact solution at the continuous level and the approximate solution that is valid at each time step and each iteration of the linearization solver. Our estimate, based on equilibrated flux reconstructions, also distinguishes the spatial and temporal components of the error. The spatial component further consists of the discretization error and the linearization error. At each time step of the simulation, we propose an adaptive version of the Newton algorithm based on stopping the Newton iterations when the estimators of the corresponding error components does not affect significantly the overall estimate. In addition, this adaptive procedure leads to computational savings while ensuring the accuracy of the solution. Numerical experiments are presented to demonstrate the effectiveness of the proposed approach.
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Bader et al. (2025) studied Cardiac electrical activity simulation. Adaptive Newton algorithm was evaluated. An adaptive version of the Newton algorithm based on a posteriori error estimates led to computational savings while ensuring solution accuracy for cardiac monodomain model simulations.
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