标题： 一种用于脑电生理学高阶近似的自适应多面体不连续伽辽金方法 显示英文标题

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

摘要： 多尺度数学模型在计算脑电生理学中显示出巨大潜力，但由于快速动力学和复杂的脑几何结构，仍受到高计算成本的阻碍，需要非常精细的空间-时间分辨率。 本文介绍了一种新颖的p自适应不连续伽辽金方法，在多面体网格（PolyDG）上结合Crank-Nicolson时间积分，以高效地近似这些模型。 p自适应方法通过由后验误差估计器引导的动态、单元自适应多项式细化/退化来增强局部精度。 一种新颖的聚类算法自动化选择用于自适应更新的单元，进一步提高了效率。 一系列数值测试，包括人脑干矢状截面的癫痫发作模拟，证明了该方法在保持数值解捕获多个波前动力学的准确性的同时，能够减少计算负载。 显示英文摘要

摘要： 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.