Laplacian transfer across a rough interface: Numerical resolution in the conformal plane

TL;DR

This paper introduces a numerical method using conformal mapping to analyze Laplacian transfer across rough interfaces, efficiently handling various boundary conditions and providing well-conditioned linear systems for simulations.

Contribution

It presents a novel conformal mapping approach that simplifies the treatment of complex boundary conditions in Laplacian transfer problems across rough interfaces.

Findings

Efficient numerical resolution of Laplacian transfer with mixed boundary conditions.

Well-conditioned linear systems derived from evanescent wave basis.

Numerical validation on rough interface models.

Abstract

We use a conformal mapping technique to study the Laplacian transfer across a rough interface. Natural Dirichlet or Von Neumann boundary condition are simply read by the conformal map. Mixed boundary condition, albeit being more complex can be efficiently treated in the conformal plane. We show in particular that an expansion of the potential on a basis of evanescent waves in the conformal plane allows to write a well-conditioned 1D linear system. These general principle are illustrated by numerical results on rough interfaces.