Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology

Abstract

Abstract. The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart. However, its numerical approximation is in general fairly expensive as a consequence of the mathematical features of this system. For this reason, a simplification of this model, called Monodomain problem is quite often adopted in order to reduce computational costs. Reliability of this model is however questionable, in particular in the presence of applied currents and in the regions where the upstroke or the late recovery of the action potential is occurring. In this paper we investigate a domain decomposition approach for this problem, where the entire computational domain is suitably split and the two models are solved in different subdomains. Since the mathematical features of the two problems are rather different, the heterogeneous coupling is non trivial. Here we investigate appropriate interface matching conditions for the coupling on non overlapping domains. Moreover, we pursue an Optimized Schwarz approach for the numerical solution of the heterogeneous problem. Convergence of the iterative method is analyzed by means of a Fourier analysis. We investigate the parameters to be selected in the matching radiation-type conditions to accelerate the convergence. Numerical results both in two and three dimensions illustrate the effectiveness of the coupling strategy.Mathematics Subject Classification. 65M55, 65N30, 92-08.

Optimized Schwarz Methods and Model Adaptivity in Electrocardiology Simulations

“…Due to a dimensional mismatch between the two models, two interface conditions are needed on the Bidomain side of the interface, and one on the Monodomain side [6]. Possible coupling conditions are …”

“…To cope with the mismatch, the second condition in (8) is a transparent boundary condition, designed to avoid spurious reflexions off the interface for the extracellular potential wave. The convergence of the Optimized Schwarz Algorithm based on the interface conditions (8)- (9) was analyzed in [6], where also optimal parameters has been identified by means of Fourier analysis.…”

“…This coupling guarantees a significant reduction of the computational costs, but raises non trivial issues when matching the two models, featuring a different size. This has been considered in [6], where the Optimized Schwarz method has been advocated for the heterogeneous coupling. From the practical standpoint, an immediate idea for setting up a model adaptive solver would be to use the a posteriori error estimator for detecting the regions where to solve the Bidomain problem and then to couple these subdomains with the Monodomain regions, so to have the Bidomain model capturing the wavefron propagation.…”

“…The Monodomain model is a simplified model for the propagation of the electrical stimulus, based upon a proportionality assumption between the intracellular and the extracellular conductivity tensors, namely assuming D e = λ D i , where λ is a constant to be properly chosen. We assume here λ = σ l e /σ l i [6], and the Monodomain model reads…”

BPX preconditioners for the Bidomain model of electrocardiology

“…The advantage of IMEX and operator splitting schemes is that they only require the solution of linear systems at each time step. Many different preconditioners have been proposed in order to devise efficient iterative solvers for such linear systems: SSOR [16], block diagonal or triangular [17,18,19,20,21,22,23], optimized Schwarz [24,25], geometric multigrid [26,27], algebraic multigrid [28,17,29,18,19], multilevel Schwarz [30,31,32], Neumann-Neumann and BDDC [33,34] preconditioners.…”