An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology

Abstract

This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in cardiac electrophysiology. The proposed approach exploits general polytopic grids at coarser levels, obtained through the agglomeration of elements from an initial, potentially fine, mesh. By leveraging a robust and efficient agglomeration strategy, we construct a nested hierarchy of grids suitable for multilevel solver frameworks. The effectiveness and performance of the methodology are assessed through a series of numerical experiments on two- and three-dimensional domains, involving different ionic models and realistic unstructured geometries. The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner.

Figures (14)

• Figure 1: (\ref{['fig:ventricle_partition']}) MPI partitioning of the realistic hexahedral ventricle mesh into $128$ subdomains. Each color corresponds to a different MPI process. Bottom row: detailed views of an agglomerate on the boundary of the domain (\ref{['fig:ventricle_agglomerate']}) and its $8$ sub-agglomerates (\ref{['fig:ventricle_sub_agglo1']}) and (\ref{['fig:ventricle_sub_agglo2']}), each displayed in a different color.
• Figure 2: MPI partitioning of $\mathcal{T}_h$
• Figure 3: $t=0.04$s
• Figure 4: $t=0.16$s
• Figure 6: Number of PCG iterations per time step for Problem \ref{['eqn:monodomain']}, comparing AMG and agglomerated multigrid (AggloMG) for polynomial degrees $p=1,2,3,4$.