A fast isogeometric solver for electrophysiology based on weak spatial form with minimal patch-wise quadrature

We present an efficient isogeometric method for solving the monodomain reaction-diffusion equation, which is fundamental in electrophysiology modeling. The Galerkin weak form is employed for spatial discretization on surfaces, enabling accurate handling of complex geometries, including multi-patch domains, and variable material properties. A semi-implicit time integration scheme is adopted to resolve the fast transients characteristic of electrophysiological dynamics. The main computational bottleneck arises from the repeated numerical integration of the forcing term coming from the ionic and applied current. To overcome this, we introduce a minimal patch-wise quadrature strategy tailored for quadratic and cubic NURBS with maximum regularity. Exploiting the high regularity of the forcing term, this scheme significantly reduces the number of integration points, approximately one per surface element, without compromising accuracy or introducing spurious solutions. The proposed solver is evaluated through several numerical experiments, ranging from planar wavefront propagation to multi-patch simulations of cardiac tissue activation considering the variable fiber orientation. The results demonstrate substantial improvements in computational efficiency, particularly for refined ionic current models governed by multiple variables, while maintaining numerical accuracy.