We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on solvers based on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that with the proper choice of subspace splitting we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regards to number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely the electric signalling in excitable tissue modeled by bidomain, EMI and reduced EMI equations.

翻译：针对界面驱动的多物理场问题，我们发展了可跨维度耦合的多层方法，其中界面耦合的复杂性和强度会降低标准方法的性能。研究重点是基于聚合型代数多重网格方法的求解器，其采用定制光滑子以在各粗层上保留耦合信息。我们证明，通过合理选择子空间分裂，可在两层框架下获得关于离散化参数和物理参数的均匀收敛性。此外，通过多个与大脑生物物理过程相关的数值算例——具体包括由双域、EMI及简化EMI方程建模的兴奋性组织电信号传导——展示了该方法在物理参数鲁棒性及系统自由度可扩展性方面的优势。