Numerical analysis of fractional-in-space FitzHugh Nagumo model with finite volume method

Used to describe the propagation of the electrical potential in heterogeneous cardiac tissue, the FitzHugh-Nagumo monodomain model has attracted numerous attentions to be analyzed. Due to the irregular profile of the cardiac tissue, there has been limitations for the numerical solution on the considered regular domains or approximate irregular domains, where the analytic solution could not be obtained. In this paper, a control volume finite element method is considered to solve the two-dimensional fractional-in-time-space FitzHugh-Nagumo monodomain model with homogeneous Neumann boundary conditions on arbitrary irregular domains with unstructured mesh. By using the linear interpolation shape function methods and matrix transfer techniques, the matrices representation of the fractional Laplacian (-∇
²
)
^α/2
are simply obtained. With a mixed explicit-implicit scheme in time, computing the discretised equations with the flux treated implicitly and the source term treated explicitly is transformed to compute matrix-function-vector products. The highlights of this method are that the considered solution domains can be arbitrary, the proposed strategy can be generalised to lower or higher dimensions and the boundary conditions can be generalised to other types. Finally, the efficiency of our method is show-cased by solving fractional-in-space FitzHugh-Nagumo models in two dimensions.