A BDDC Preconditioner for the Cardiac EMI Model in three Dimensions

TL;DR

This paper develops and analyzes a BDDC preconditioner tailored for three-dimensional composite Discontinuous Galerkin discretizations of cardiac reaction-diffusion models, enhancing computational efficiency and scalability.

Contribution

It introduces a novel BDDC preconditioner specifically designed for complex 3D cardiac EMI models with discontinuous solutions, providing theoretical bounds and validation.

Findings

Scalable condition number bound established

Numerical experiments confirm theoretical predictions

Improved solution efficiency for cardiac 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.