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Optimal error estimates of the diffuse domain method for parabolic equations

Wenrui Hao, Lili Ju, Yuejin Xu

TL;DR

The paper addresses solving parabolic PDEs with Neumann boundary on irregular domains using the diffuse domain method (DDM). By introducing a phase-field weight $\omega_{\epsilon}$ to extend the problem to a larger rectangular domain and analyzing the limit $\epsilon\to 0$, it obtains rigorous convergence and optimal error estimates in weighted $L^2$ and $H^1$ norms: $\|u^{\epsilon}-u\|_{L^2(D_{\epsilon};\omega_{\epsilon})}\lesssim \epsilon^2$ and $\|u^{\epsilon}-u\|_{H^1(D_{\epsilon};\omega_{\epsilon})}\lesssim \epsilon$. The results are supported by numerical experiments for both constant and varying diffusion coefficients, confirming the predicted rates and demonstrating the method's effectiveness on complex geometries. This work provides the first rigorous parabolic DDM analysis and offers a framework that decouples constants from the interface thickness, with potential implications for adaptive finite element methods on irregular domains.

Abstract

In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a similar but slightly modified problem defined over a larger rectangular domain that contains the target physical domain. Based on the weighted Sobolev spaces, we rigorously establish the convergence of the diffuse domain solution to the original solution as the interface thickness parameter goes to zero, together with the corresponding optimal error estimates under the weighted $L^2$ and $H^1$ norms. Numerical experiments are also presented to validate the theoretical results.

Optimal error estimates of the diffuse domain method for parabolic equations

TL;DR

The paper addresses solving parabolic PDEs with Neumann boundary on irregular domains using the diffuse domain method (DDM). By introducing a phase-field weight to extend the problem to a larger rectangular domain and analyzing the limit , it obtains rigorous convergence and optimal error estimates in weighted and norms: and . The results are supported by numerical experiments for both constant and varying diffusion coefficients, confirming the predicted rates and demonstrating the method's effectiveness on complex geometries. This work provides the first rigorous parabolic DDM analysis and offers a framework that decouples constants from the interface thickness, with potential implications for adaptive finite element methods on irregular domains.

Abstract

In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a similar but slightly modified problem defined over a larger rectangular domain that contains the target physical domain. Based on the weighted Sobolev spaces, we rigorously establish the convergence of the diffuse domain solution to the original solution as the interface thickness parameter goes to zero, together with the corresponding optimal error estimates under the weighted and norms. Numerical experiments are also presented to validate the theoretical results.

Paper Structure

This paper contains 12 sections, 10 theorems, 82 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The diffuse domain method
  3. Preliminaries
  4. On the weighted Sobolev space
  5. On diffuse volume integrals
  6. Error analysis
  7. Optimal error estimate in the $L^2$-norm
  8. Optimal error estimate in the $H^1$ norm
  9. Numerical experiments
  10. The case of constant diffusion coefficient
  11. The case of varying diffusion coefficient
  12. Conclusions

Key Result

Theorem 1

Let $\epsilon_0>0$ be sufficiently small and $1\leq p < \infty$. Then, there exists a constant $C>0$ such that for any $\epsilon \in [0, \epsilon_0]$ and $v \in W^{1,p}(D_{\epsilon};\omega_{\epsilon})$, there holds

Figures (3)

  • Figure 1: The relationship among $D$, $D_{\epsilon}$ and $\Omega$ for some $\epsilon>0$.
  • Figure 2: The phase structures of the numerical solutions at the terminal time $T=0.5$ produced by the DDM approach with interface thickness $\epsilon=1/16,1/32,1/64$ (from left to right) for Example \ref{['ex1']} in the circular domain (top row) and the flower-shaped domain (bottom row).
  • Figure 3: The phase structures of the numerical errors at the terminal time $T=0.5$ produced by the DDM approach with interface thickness $\epsilon=1/16,1/32,1/64$ (from left to right) for Example \ref{['ex2']} in the circular domain (top row) and the flower-shaped domain (bottom row).

Theorems & Definitions (17)

  • Theorem 1: Trace Theorem
  • Theorem 2: Embedding Theorem
  • Theorem 3: Poincare-Friedrichs-type inequality
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • Theorem 6: Error estimate in the $L^2$ norm
  • proof
  • Lemma 2
  • ...and 7 more