  The bidomain equations have been widely used to mathematically model the
electrical activity of the cardiac tissue. In this work, we present a potential
theory-based Cartesian grid method which is referred as the kernel-free
boundary integral (KFBI) method which works well on complex domains to
efficiently simulate the linear diffusion part of the bidomain equation. After
a proper temporal discretization, the KFBI method is applied to solve the
resulting homogeneous Neumann boundary value problems with a second-order
accuracy. According to the potential theory, the boundary integral equations
reformulated from the boundary value problems can be solved iteratively with
the simple Richardson iteration or the Krylov subspace iteration method. During
the iteration, the boundary and volume integrals are evaluated by limiting the
structured grid-based discrete solutions of the equivalent interface problems
at quasi-uniform interface nodes without the need to know the analytical
expression of Green's functions. In particular, the discrete linear system of
the equivalent interface problem obtained from the standard finite difference
schemes or the finite element schemes can be efficiently solved by fast
elliptic solvers such as the fast Fourier transform based solvers or those
based on geometric multigrid iterations after an appropriate modification at
the irregular grid nodes. Numerical results for solving the FitzHugh-Nagumo
bidomain equations in both two- and three-dimensional spaces are presented to
demonstrate the numerical performance of the KFBI method such as the
second-order accuracy and the propagation and scroll wave of the voltage
simulated on the real human left ventricle model.
