Solving PDEs in irregular geometries with multiresolution methods I: Embedded Dirichlet boundary conditions
In this work, we develop and analyze a formalism for solving boundary value problems in arbitrarily-shaped domains using the MADNESS (multiresolution adaptive numerical environment for scientific simulation) package for adaptive computation with multiresolution algorithms. We begin by implementing a previously-reported diffuse domain approximation for embedding the domain of interest into a larger domain (Li et al., 2009 [1]). Numerical and analytical tests both demonstrate that this approximation yields non-physical solutions with zero first and second derivatives at the boundary. This excessive smoothness leads to large numerical cancellation and confounds the dynamically-adaptive, multiresolution algorithms inside MADNESS. We thus generalize the diffuse domain approximation, producing a formalism that demonstrates first-order convergence in both near- and far-field errors. We finally apply our formalism to an electrostatics problem from nanoscience with characteristic length scales ranging from 0.0001 to 300 nm. (C) 2011 Elsevier B.V. All rights reserved.
electrostatics multiresolution analysis domain embedding techniques