Speaker
Description
Traveling wave-like phenomena arise in a wide range of biological processes, including electrical signal propagation in the nervous system and brain tissue. Accurately simulating these phenomena poses significant numerical challenges, as sharp and rapidly moving wavefronts require high spatial and temporal resolution to be properly captured. Such requirements often lead to prohibitive computational costs, especially in large and heterogeneous domains. The evolution of the transmembrane potential is governed by steep, fast wavefronts propagating across multiple brain regions, often following preferential axonal pathways. These dynamics emerge from complex multiscale interactions, where rapid ionic exchanges at the cellular level generate electrical signals that propagate through anisotropic and heterogeneous tissues. High-order discontinuous Galerkin methods on polygonal and polyhedral meshes (PolyDG) offer great flexibility and accuracy for such problems, but their computational cost can remain significant when uniform high-order discretizations are used. To address this limitation, we propose a p-adaptive PolyDG strategy that exploits the solution's intrinsic traveling-wave structure. In particular, the transmembrane potential exhibits strong spatial variations localized near the propagating wavefront, while the solution remains nearly stationary elsewhere. Our approach consists of designing efficient a posteriori error indicators that accurately detect the wavefront location. These indicators drive fully automatic local adjustment of the polynomial degree, allowing higher-order approximations only where they are truly needed. We present numerical results that validate the effectiveness of the proposed method and demonstrate its application to the simulation of epileptic events in heterogeneous brain domains, including grey and white matter. The results show that polynomial adaptivity significantly reduces the total number of degrees of freedom and the overall computational cost while maintaining high-order accuracy throughout the simulation.