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A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence

Toai Luong, Tadele Mengesha, Steven M. Wise, Ming Hei Wong

TL;DR

The paper analyzes a diffuse-domain method for PDEs on domains with interfaces by introducing a phase-field $\phi_\varepsilon$ and studying the energy functionals $F_\varepsilon$ that encode the diffuse problem. It proves that, for the Neumann case ($\kappa=0$), $F_\varepsilon$ $\Gamma$-converges to the sharp-interface functional $F_0$ as $\varepsilon\to0$ in any dimension, and that minimizers $u_\varepsilon$ converge strongly to the sharp-interface minimizer $u_0$ in $H^1(\Omega)$ up to subsequences. In 1D, the results extend to general Robin-type transmission ($\kappa\ge0$), with the interfacial term converging to the boundary contribution and strong $H^1(\Omega)$ convergence still holding. The analysis relies on weighted diffuse-interface lemmas, compactness, and explicit liminf/limsup constructions, providing a rigorous justification of the diffuse-domain approach for two-sided transmission problems. The findings support reliable numerical and analytical use of DDMs in complex geometries and highlight dimensional nuances, especially in 1D where the Robin case is tractable.

Abstract

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This reformulation extends the problem to a regular domain, incorporating boundary conditions via singular source terms. In this work, we analyze the convergence of a DDM approximation problem with transmission-type Neumann boundary conditions. We prove that the energy functional of the diffuse domain problem $Γ$--converges to the energy functional of the original problem as $\varepsilon \to 0$. Additionally, we show that the solution of the diffuse domain problem strongly converges in $H^1(Ω)$, up to a subsequence, to the solution of the original problem, as $\varepsilon \to 0$.

A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence

TL;DR

The paper analyzes a diffuse-domain method for PDEs on domains with interfaces by introducing a phase-field and studying the energy functionals that encode the diffuse problem. It proves that, for the Neumann case (), -converges to the sharp-interface functional as in any dimension, and that minimizers converge strongly to the sharp-interface minimizer in up to subsequences. In 1D, the results extend to general Robin-type transmission (), with the interfacial term converging to the boundary contribution and strong convergence still holding. The analysis relies on weighted diffuse-interface lemmas, compactness, and explicit liminf/limsup constructions, providing a rigorous justification of the diffuse-domain approach for two-sided transmission problems. The findings support reliable numerical and analytical use of DDMs in complex geometries and highlight dimensional nuances, especially in 1D where the Robin case is tractable.

Abstract

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness , which scales with the minimum grid size. This reformulation extends the problem to a regular domain, incorporating boundary conditions via singular source terms. In this work, we analyze the convergence of a DDM approximation problem with transmission-type Neumann boundary conditions. We prove that the energy functional of the diffuse domain problem --converges to the energy functional of the original problem as . Additionally, we show that the solution of the diffuse domain problem strongly converges in , up to a subsequence, to the solution of the original problem, as .

Paper Structure

This paper contains 14 sections, 9 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. $\Gamma$--convergence of the Energy Functional for the Neumann Boundary Condition case
  4. Strong $H^1(\Omega)$--convergence of the Approximation Solution for the Neumann Boundary Condition case
  5. Proof of Theorem \ref{['thm-main-2']}: $\Gamma$--convergence of ${\@fontswitch{}{\mathcal{}} F}_\varepsilon$
  6. Preliminary Lemmas
  7. Compactness
  8. Liminf Inequality
  9. Limsup Inequality and Recovery Sequence
  10. Proof of Theorem \ref{['thm-main-3']}: Strong convergence of $u_\varepsilon$ in $H^1(\Omega)$
  11. $\Gamma$-convergence and Strong $H^1(\Omega)$--convergence in 1D
  12. $\Gamma$--convergence of the Energy Functional in 1D
  13. Strong $H^1(\Omega)$--convergence of the Approximation Solution in 1D
  14. Discussion

Key Result

Theorem 2.1

\newlabelthm-main-2 As $\varepsilon \to 0$, ${\@fontswitch{}{\mathcal{}} F}_\varepsilon$$\Gamma$--converges to ${\@fontswitch{}{\mathcal{}} F}_0$ under the strong $L^2(\Omega)$--topology.

Figures (1)

  • Figure 1.1: A domain $\Omega_1$ is covered by a larger cuboidal domain $\Omega$. $\Omega_2 := \Omega \setminus \overline{\Omega_1}$.

Theorems & Definitions (17)

  • Theorem 2.1: $\Gamma$--convergence of ${\@fontswitch{}{\mathcal{}} F}_\varepsilon$
  • Theorem 2.2: Strong convergence of $u_\varepsilon$ in $H^1(\Omega)$
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Remark 3.1
  • Theorem 3.5: Compactness
  • ...and 7 more