A discrete de Rham discretization of interface diffusion problems with application to the Leaky Dielectric Model

TL;DR

This paper introduces a high-order discrete de Rham scheme for interface elliptic problems, effectively handling potential and flux jumps, with applications to electrodynamics models like the Leaky Dielectric Model.

Contribution

It develops a novel arbitrary-order discretization method that supports complex interface geometries and enforces conditions weakly, with proven convergence and practical application.

Findings

Rigorous convergence analysis for the proposed scheme.

Effective handling of interface jumps in potential and flux.

Successful application to a physical electrodynamics problem.

Abstract

Motivated by the study of the electrodynamics of particles, we propose in this work an arbitrary-order discrete de Rham scheme for the treatment of elliptic problems with potential and flux jumps across a fixed interface. The scheme seamlessly supports general elements resulting from the cutting of a background mesh along the interface. Interface conditions are enforced weakly \`a la Nitsche. We provide a rigorous convergence of analysis of the scheme for a steady model problem and showcase an application to a physical problem inspired by the Leaky Dielectric Model.