Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics

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

Abstract

Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This approach reformulates the original equations on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM for a two-sided problem with transmission-type Robin boundary conditions. Our results show that, in the one dimensional space, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in $\varepsilon$. Furthermore, we provide numerical simulations that validate and illustrate the analytical result.

A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics

Abstract

Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness , which scales with the minimum grid size. This approach reformulates the original equations on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM for a two-sided problem with transmission-type Robin boundary conditions. Our results show that, in the one dimensional space, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in . Furthermore, we provide numerical simulations that validate and illustrate the analytical result.

Paper Structure

This paper contains 17 sections, 2 theorems, 184 equations, 21 figures.

Table of Contents

  1. Introduction
  2. The One-sided Diffuse Domain Problem
  3. The Two-sided Diffuse Domain Problem
  4. A Singular $\alpha \to 0$ Limit of the Two-Sided Problem
  5. Main Result
  6. Asymptotic Analysis for the Diffuse Domain Problem in 1D
  7. Matched Asymptotics for Corner Layer Problems
  8. Outer expansions
  9. Supporting Lemmas
  10. Inner expansions
  11. Equation (\ref{['1D-diff-bvp1-inner']}) at $O(1)$
  12. Equation (\ref{['1D-diff-bvp1-inner']}) at $O(\varepsilon)$
  13. Equation (\ref{['1D-diff-bvp1-inner']}) at $O(\varepsilon^2)$
  14. Numerical Simulations
  15. Numerical Simulations in 1D
  16. ...and 2 more sections

Key Result

Lemma 3.1

\newlabelrational-lemma If $F(z) = P(z) + o(1)$ and $G(z) = A + o(z^{-m})$ as $z \to \infty$ (or $z \to -\infty$), where $P(z)$ is a polynomial, $A \neq 0$ is a constant, and $m > \deg(P)$ is an integer, then

Figures (21)

  • Figure 1.1: A domain $\Omega_1$ is covered by a larger cuboidal domain $\Omega$, with $\Omega_2 := \Omega \setminus \overline{\Omega_1}$.
  • Figure 3.1: The domains $\Omega = (-1,1)$, $\Omega_L = (-1,0)$, $\Omega_R = (0,1)$, and the graph of $\phi_\varepsilon (x)$ in 1D.
  • Figure 3.2: A sketch of the regions used for the matched asymptotic expansions.
  • Figure 3.3: An example of a corner layer. The translucent blue curve is the zeroth-order outer approximation, $u_0$, and the translucent red curve represents the zeroth-order inner approximation, $U_0$. The solid black line is the diffuse-domain solution $u_\varepsilon(x)$. Not shown is the composite approximation, $u_{c,0}(x;\varepsilon)$, obtained by combining $u_0$ and $U_0$ via a process known as matching. See estimate \ref{['eqn:comp-approx']} and associated discussion.
  • Figure 4.1: Plots of the true solution $u_0$ specified by (\ref{['true-sln-1D']}) and the diffuse domain approximation solution $u_\varepsilon$ over $\Omega = (-1,1)$, for various values of $\varepsilon$, with the parameters given in (\ref{['data1D-1']})--(\ref{['data1D-2']}).
  • ...and 16 more figures

Theorems & Definitions (8)

  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.2