Abstract
Many problems in Biology and Engineering are
governed by anisotropic reactiondiffusion 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 reactiondiffusion 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.
