High-order Discontinuous Galerkin Methods for the Monodomain and Bidomain Models

TL;DR

This work develops a high-order discontinuous Galerkin method with a spectral (Dubiner) basis to discretize the monodomain and bidomain cardiac electrophysiology models, integrating a semi-implicit time stepping scheme to efficiently advance solutions. It provides a complete semi-discrete and fully discrete framework, including the bilinear forms, stabilization, and algebraic assembly, and demonstrates convergence orders consistent with theoretical expectations for both models. The approach successfully reproduces key physiological phases, such as depolarization and repolarization, even on anisotropic meshes, and includes a pseudo-realistic 2D simulation driven by a localized external current. Collectively, the results indicate that high-order DG with spectral bases can deliver accurate, scalable simulations of cardiac electrical propagation on practical grids, with potential implications for faster, high-fidelity cardiac simulations.

Abstract

This work aims at presenting a Discontinuous Galerkin (DG) formulation employing a spectral basis for two important models employed in cardiac electrophysiology, namely the monodomain and bidomain models. The use of DG methods is motivated by the characteristic of the mathematical solution of such equations which often corresponds to a highly steep wavefront. Hence, the built-in flexibility of discontinuous methods in developing adaptive approaches, combined with the high-order accuracy, can well represent the underlying physics. The choice of a semi-implicit time integration allows for a fast solution at each time step. The article includes some numerical tests to verify the convergence properties and the physiological behaviour of the numerical solution. Also, a pseudo-realistic simulation turns out to fully reconstruct the propagation of the electric potential, comprising the phases of depolarization and repolarization, by overcoming the typical issues related to the steepness of the wave front.

Figures (6)

• Figure 1: Color plot of the transformation (\ref{['transformation_formula']}) underlying the construction of the 2D Dubiner spectral basis on simplices. The collapse of the square can be seen on the upper edge of the triangle.
• Figure 2: Contour plot of the exact solution for $\phi_e$ in \ref{['sec:verificationtestcases']}.
• Figure 3: Sample of uniformly refined grids with granularity $h\approx 2^{-\sigma}$, $\sigma=0,1,2$.
• Figure 4: Computed errors vs mesh size (log-log scale) for $p=1,2$: monodomain (top) and bidomain (bottom) models.
• Figure 5: Computed errors vs polynomial degree $p$ (semilogy scale) for monodomain model, $h\approx 2^{-3}$.

Theorems & Definitions (4)

• Definition 1: Jacobi polynomials jacobi
• Definition 2: 2D Dubiner basis functions antonietti2011
• Remark 1
• Remark 2