A BDDC Preconditioner for the Cardiac EMI Model in three Dimensions

TL;DR

This work extends a previously developed Balancing Domain Decomposition by Constraints (BDDC) framework to three-dimensional Discontinuous Galerkin discretizations of the EMI cardiac model, enabling efficient solution of cell-by-cell micro-models. It derives a two-level BDDC preconditioner based on carefully crafted dual and primal spaces and a rho-scaled jump operator, proving a polylogarithmic condition-number bound independent of conductivity coefficients. The authors validate the theory with scalable numerical experiments on 3D geometries, demonstrating robust performance under mesh refinement and parameter variation, and discuss the implications for large-scale cardiac simulations. The results offer a scalable, coefficient-robust solver for EMI DG models, paving the way for integrative, high-fidelity cardiac simulations beyond homogenized macroscopic models.

Abstract

We analyze a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for the solution of three dimensional composite Discontinuous Galerkin discretizations of reaction-diffusion systems of ordinary and partial differential equations arising in cardiac cell-by-cell models like the Extracellular space, Membrane and Intracellular space (EMI) Model. These microscopic models are essential for the understanding of events in aging and structurally diseased hearts which macroscopic models relying on homogenized descriptions of the cardiac tissue, like Monodomain and Bidomain models, fail to adequately represent. The modeling of each individual cardiac cell results in discontinuous global solutions across cell boundaries, requiring the careful construction of dual and primal spaces for the BDDC preconditioner. We provide a scalable condition number bound for the precondition operator and validate the theoretical results with extensive numerical experiments.

Figures (9)

• Figure 1: Visualization of two cells $\Omega_1$ (green) and $\Omega_2$ (blue) floating in extracellular liquid $\Omega_0$ (grey). For this three-dimensional example, we have to consider interface terms for ionic currents between extracellular space and the cells over $F_{01} = \partial\Omega_0 \cap \partial\Omega_1$ and $F_{02} = \partial \Omega_0 \cap \partial \Omega_2$ and ionic currents through gap junctions between the cells over $F_{12} = \partial\Omega_1\cap\partial\Omega_2$. For the BDDC preconditioner, it is important to also consider edge terms on $E_{0, \{1, 2\}} = \partial\Omega_0\cap\partial\Omega_1\cap\partial\Omega_2$.
• Figure 2: Schematic visualization of the FE space for a tetrahedral substructure $\Omega_i$ (grey) with two neighboring substructures. $u_i \in W(\Omega_i')$ will consist of $(u_i)_i$ on the substructure $\Omega_i$ and of $(u_i)_j$ and $(u_i)_k$, the traces of $V_{\{j, k\}}(\Omega_{\{j, k\}})$ on the faces $F_{ji}$ (green) and $F_{ki}$ (blue), respectively.
• Figure 3: Representation of primal constraints connected to a tetrahedral substructure $\Omega_i$ (grey) surrounded by three neighbors (red, green and blue), with the fourth face intersecting with $\partial \Omega$ (the global Neumann boundary). For each face, edge and vertex, one primal constraint per involved substructure is created and on each of them, the according averaging constraints are imposed. In this particular example, $\Omega_i$ will contribute to the primal space with 6 face average, 9 edge average and 4 vertex constraints.
• Figure 4: Repetitive test geometry with 3x3x1 (left), 3x3x3 (middle) and 4x4x4 (right) cells. Each cell is an intracellular subdomain inside a cube that can be stacked in all dimensions, resulting in a mesh where all intracellular subdomains have interfaces with extracellular space via a cell membrane model and with other intracellular domains via a linear gap junction.
• Figure 5: Iterations needed to converge to a relative residual tolerance of $10^{-6}$ (left) and condition number estimates (right) for random right-hand side vectors. The results colored in blue consider a full primal space containing vertex values as well as edge and face averages. The results colored in red consider only vertex values and edge averages in the primal space. The solid lines show the mean over 100 different random right-hand sides, the colored areas represent the range of iterations or condition numbers for each test case, respectively.

Theorems & Definitions (18)

• Remark 1
• Definition 1
• Definition 2: Subspaces $\widetilde{W}(\Omega')$ and $\widetilde{W}(\Gamma')$
• Definition 3: Subspaces $\widetilde{W}_{VE}(\Omega')$ and $\widetilde{W}_{VE}(\Gamma')$
• Lemma 1
• Remark 2
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5