The Smoothed Boundary method is a recently proposed fictitious
domain method to approximate the solution of partial differential
equations in irregular domains with no-flux (homogeneous Neumann)
boundary conditions. The method relies in embedding the irregular
geometry in a slightly bigger rectangular domain and generating an
auxiliary function that is a smoothed regular approximation to the
characteristic function that defines the irregular domain. This
auxiliary function is then suitably introduced in the original
problem letting to the formulation of a new problem whose solution
converges to the solution of the original partial differential
equation and that incorporates the desired boundary conditions
over the irregular boundary. Since the new extended problem can be
discretized in space using any standard numerical stencil, our
idea also opens the door to spectral methods to be considered as
potent solvers for the solution of problems in complex geometries.

In this seminar, we focus on presenting the main ideas of the
method and its application to the solution of some parabolic
problems, with special interest in its use for the simulation of
cardiac electrical excitation in three-dimensional realistic
ventricular geometries of the heart.