High-order Spectral/ Element Discretisation for Reaction-diffusion Problems on Surfaces: Application to Cardiac Electrophysiology

We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/ elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the computational cost of anatomically-realistic models can be significantly reduced through the surface representation and use of high-order methods. We describe and validate our discretisation and provide a demonstration of its application to modelling electrochemical propagation across a human left atrium.