GDSW preconditioners for composite Discontinuous Galerkin discretizations of multicompartment reaction-diffusion problems

TL;DR

This paper introduces a generalized GDSW preconditioner for composite Discontinuous Galerkin methods applied to multicompartment reaction-diffusion problems, demonstrating theoretical scalability and numerical robustness in cardiac bioelectric models.

Contribution

The work develops and analyzes a new GDSW preconditioner tailored for complex reaction-diffusion systems with discontinuities, showing its effectiveness and scalability.

Findings

Preconditioned operator converges with scalable, quasi-optimal bounds.

Numerical tests confirm robustness against time step size and coefficient jumps.

Preconditioner effectively accelerates solution of cardiac bioelectric models.

Abstract

The aim of the present work is to design, analyze theoretically, and test numerically, a generalized Dryja-Smith-Widlund (GDSW) preconditioner for composite Discontinuous Galerkin discretizations of multicompartment parabolic reaction-diffusion equations, where the solution can exhibit natural discontinuities across the domain. We prove that the resulting preconditioned operator for the solution of the discrete system arising at each time step converges with a scalable and quasi-optimal upper bound for the condition number. The GDSW preconditioner is then applied to the EMI (Extracellular - Membrane - Intracellular) reaction-diffusion system, recently proposed to model microscopically the spatiotemporal evolution of cardiac bioelectrical potentials. Numerical tests validate the scalability and quasi-optimality of the EMI-GDSW preconditioner, and investigate its robustness with respect to the time step size as well as jumps in the diffusion coefficients.