The bidomain equation is the most commonly used model to describe in detail the spatial and temporal electric activity of the heart. The solutions are often travelling waves characterised by a very steep front, about 0.1 mm thick. This yields a considerable computational effort on patient-specific human geometries.

The first aim of this work is to analyse in detail the effect of a specific discretisation in space on traveling wave solutions of the bistable (or Nagumo) equation, which is a simple but mathematically reasonable approximation of the dynamics at the front of the wave. We show how specific discretisation schemes (FD, FE, DG), and the choice of a coarse grid, can affect the solution, leading so to possibly erroneous physiological conclusions.

In this talk we provide several error estimates of the conduction velocity for different numerical schemes, by performing a perturbation analysis of the discrete problem associated to the discrete traveling wave. We also analyse the impact of mass lumping, adopted by several authors in relation with operator splitting schemes.

Secondly, we exploit the error estimates to design a robust numerical scheme for the problem of interest. We propose and analyse two different schemes: the first one is a fourth-order (in space) scheme obtained by a weighted average of the finite difference and the finite element method. The second one is a “stabilised” finite element scheme, where we introduce a numerical conductivity to consistently adjust the conduction velocity
These novel schemes show very good approximation properties of the conduction velocity.

Finally, we test the methodology on realistic ionic models and geometries. The results are in excellent agreement with our analytical results based on the bistable model, thus showing the validity of the approach.

This is a joint work with Johan Hake and Joakim Sundnes.