Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Daniele A. Di Pietro, Simon Mendez, Aurelio Edoardo Spadotto

TL;DR

This work develops a high-order discrete de Rham discretization for elliptic interface problems with potential and flux jumps, allowing unfitted cuts of the interface on general polytopal meshes. Interface conditions are enforced weakly via Nitsche-type terms and trace reconstructions, with diffusion-weighted averages ensuring robustness to large coefficient contrasts. The authors prove stability and optimal-order error estimates in an energy-like norm and demonstrate robust, convergent behavior on square, circular, and generic interfaces, including curved interfaces. The method is extended to a time-dependent Leaky Dielectric Model with a capacitive interface jump, and numerical tests validate both spatial and temporal convergence, highlighting the framework’s applicability to moving-interface electrohydrodynamics.

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 à 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.

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

TL;DR

This work develops a high-order discrete de Rham discretization for elliptic interface problems with potential and flux jumps, allowing unfitted cuts of the interface on general polytopal meshes. Interface conditions are enforced weakly via Nitsche-type terms and trace reconstructions, with diffusion-weighted averages ensuring robustness to large coefficient contrasts. The authors prove stability and optimal-order error estimates in an energy-like norm and demonstrate robust, convergent behavior on square, circular, and generic interfaces, including curved interfaces. The method is extended to a time-dependent Leaky Dielectric Model with a capacitive interface jump, and numerical tests validate both spatial and temporal convergence, highlighting the framework’s applicability to moving-interface electrohydrodynamics.

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 à 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.
Paper Structure (23 sections, 5 theorems, 63 equations, 13 figures)

This paper contains 23 sections, 5 theorems, 63 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Continuous setting
  3. Discrete setting
  4. Mesh
  5. Local polynomial spaces
  6. Discrete problem
  7. Discrete space
  8. Element gradient and potential
  9. Interface trace operators
  10. Discrete problem
  11. Energy norm and interface jump seminorm
  12. Main results
  13. Numerical tests
  14. Square interface
  15. Circular interface
  16. ...and 8 more sections

Key Result

Proposition 6

The map $\| \cdot \|_{{\rm en},h}$ defines a norm on $\underline{V}_{h,0}^k$.

Figures (13)

  • Figure 1: Configuration for the continuous problem.
  • Figure 2: The exact solution \ref{['eq:square.test:exact.solution']} considered in Section \ref{['sec:numerical.tests:square']} and its gradient for $\frac{\sigma_\mathrm{int}}{\sigma_\mathrm{ext}} = 10^{-1}$
  • Figure 3: Mesh sequences considered in the numerical test of Section \ref{['sec:numerical.tests:square']}.
  • Figure 4: Convergence for different values of $\frac{\sigma_\mathrm{int}}{\sigma_\mathrm{ext}}$ over a mesh sequence of Cartesian orthogonal meshes, as described in Section \ref{['sec:numerical.tests:square']}. Error is normalized with respect to the norm of the reference solution.
  • Figure 5: Convergence for different values of $\frac{\sigma_\mathrm{int}}{\sigma_\mathrm{ext}}$ over a mesh sequence of irregular quadrilaterals as described in Section \ref{['sec:numerical.tests:square']}. Error is normalized with respect to the norm of the reference solution
  • ...and 8 more figures

Theorems & Definitions (18)

  • Remark 1: Fitted mesh
  • Remark 2: Approximation of the interface
  • Remark 3: Edge potential
  • Remark 4: Validity of \ref{['eq:pT']}
  • Remark 5: Formulation of the interface terms
  • Proposition 6: Energy norm
  • proof
  • Lemma 7: Stability
  • proof
  • Theorem 8: Error estimate
  • ...and 8 more