A p-adaptive polytopal discontinuous Galerkin method for high-order approximation of brain electrophysiology

TL;DR

A novel p-adaptive discontinuous Galerkin method on polytopal grids (PolyDG) coupled with Crank-Nicolson time integration to approximate such models efficiently and demonstrates the method's ability to reduce computational load while maintaining the accuracy of the numerical solution in capturing the dynamics of multiple wavefronts.

Abstract

Multiscale mathematical models have shown great promise in computational brain electrophysiology but are still hindered by high computational costs due to fast dynamics and complex brain geometries, requiring very fine spatio-temporal resolution. This paper introduces a novel p-adaptive discontinuous Galerkin method on polytopal grids (PolyDG) coupled with Crank-Nicolson time integration to approximate such models efficiently. The p-adaptive method enhances local accuracy via dynamic, element-wise polynomial refinement/de-refinement guided by a-posteriori error estimators. A novel clustering algorithm automatizes the selection of elements for adaptive updates, further improving efficiency. A wide set of numerical tests, including epileptic seizure simulations in a sagittal section of a human brain stem, demonstrate the method's ability to reduce computational load while maintaining the accuracy of the numerical solution in capturing the dynamics of multiple wavefronts.