Solving PDEs in Complex Geometries with Multiresolution Methods: Embedded Dirichlet Boundary Conditions

Matthew Reuter, Northwestern University

We analyze the near- and far-field errors arising from a recently-proposed diffuse domain approximation for boundary value problems embedded in larger domains. We demonstrate both analytically and numerically that, although the error at the boundary is cubic with the width of the diffuse layer, the error away from the boundary converges only linearly. Moreover, the solution at the boundary has non-physical, vanishing first and second derivatives, which gives rise to computational problems in dynamically adaptive, multiresolution
algorithms. We propose and evaluate, using the MADNESS package for adaptive computation with multiresolution algorithms, modifications that lead to global
first-order convergence with improved physical and computational characteristics. Finally, we demonstrate these modifications with an electrostatics example from nanophotonics that spans several length scales.

Abstract Author(s): M. G. Reuter, J. C. Hill, and R. J. Harrison