Algebraic multigrid methods for metric-perturbed coupled problems

TL;DR

This paper introduces algebraic multigrid methods tailored for complex coupled multiphysics problems, ensuring robust and scalable solutions across different physical parameters and discretizations.

Contribution

It develops aggregation-based algebraic multigrid solvers with custom smoothers that preserve coupling, proving uniform convergence and robustness for interface-driven multiphysics problems.

Findings

Achieves uniform convergence across discretization and physical parameters.

Demonstrates robustness and scalability in biophysical brain models.

Provides numerical evidence of solver efficiency for complex coupled systems.

Abstract

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.