Multilevel Schwarz and Multigrid Preconditioners for the Bidomain System

Summary. Two parallel and scalable multilevel preconditioners for the Bidomain system in computational electrocardiology are introduced and studied. The Bidomain system, consisting of two degenerate parabolic reaction-diffusion equations coupled with a stiff system of several ordinary differential equations, generates very ill-conditioned discrete systems when discretized with semi-implicit methods in time and finite elements in space. The multilevel preconditioners presented in this paper attain the best performance to date, both in terms of convergence rate and solution time and outperform the simpler one-level preconditioners previously introduced. Parallel numerical results, using the PETSc library and run on Linux Clusters, show the scalability of the proposed preconditioners and their efficiency on largescale simulations of a complete cardiac cycle.

Adaptive Macro Finite Elements for the Numerical Solution of Monodomain Equations in Cardiac Electrophysiology

[EN] Many problems in Biology and Engineering are governed by anisotropic reaction diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.Heidenreich, E.; Ferrero De Loma-Osorio, JM.; Doblaré Castellano, M.; Rodríguez Matas, JF. (2010). 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