Smoothed Boundary Method for Diffusion-Related Partial Differential Equations in Complex Geometries
S. O. Poulsen, Peter W. Voorhees · International Journal of Computational Methods · 2017
The smoothed boundary method has emerged as an attractive method for solving partial differential equations in domains of complex morphology. In the current work, we present two schemes for incorporating Dirichlet boundary conditions on complex domains into partial differential equations through an energy penalty formulation of the smoothed boundary method, which may be straightforwardly combined with, e.g., variational phase-field methods. Convergence to the exact solution is guaranteed in a certain limit, and it is shown how parameters of the smoothed boundary method should be chosen to minimize error, and avoid additional error introduced by poor spatial resolution during numerical solution. An example demonstrating the simplicity of the implementation of the methods is presented, and the scheme believed to have most general merit is determined.