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Optimal error estimates of the diffuse domain method for semilinear parabolic equations

Yuejin Xu

TL;DR

The paper develops a rigorous error framework for the diffuse domain method (DDM) applied to semilinear parabolic equations with Neumann boundary on irregular domains. By leveraging a phase-field weight and weighted Sobolev spaces, it proves convergence of the DDM solution to the exact solution as the interface width $\epsilon$ vanishes and derives optimal $L^2$ and $H^1$ error rates: $O(\epsilon^{2})$ in $L^2$ and $O(\epsilon)$ in $H^1$. These results are complemented by numerical experiments on constant and varying diffusion coefficients and an Allen–Cahn model, confirming the theoretical rates and the method’s robustness on complex geometries. Overall, the work advances reliable, mesh-agnostic numerical treatment of PDEs on irregular domains and lays groundwork for adaptive discretizations in challenging geometries.

Abstract

In this paper, we mainly discuss the convergence behavior of diffuse domain method (DDM) for solving semilinear parabolic equations with Neumann boundary condition defined in general irregular domains. We use a phasefield function to approximate the irregular domain and when the interface thickness tends to zero, the phasefield function will converge to indicator function of the original domain. With this function, we can modify the problem to another one defined on a larger rectangular domain that contains the targer physical domain. Based on the weighted Sobolev spaces, we prove that when the interface thickness parameter goes to zero, the numerical solution will converge to the exact solution. Also, we derive the corresponding optimal error estimates under the weighted L2 and H1 norms. Some numerical experiments are also carried out to validate the theoretical results.

Optimal error estimates of the diffuse domain method for semilinear parabolic equations

TL;DR

The paper develops a rigorous error framework for the diffuse domain method (DDM) applied to semilinear parabolic equations with Neumann boundary on irregular domains. By leveraging a phase-field weight and weighted Sobolev spaces, it proves convergence of the DDM solution to the exact solution as the interface width vanishes and derives optimal and error rates: in and in . These results are complemented by numerical experiments on constant and varying diffusion coefficients and an Allen–Cahn model, confirming the theoretical rates and the method’s robustness on complex geometries. Overall, the work advances reliable, mesh-agnostic numerical treatment of PDEs on irregular domains and lays groundwork for adaptive discretizations in challenging geometries.

Abstract

In this paper, we mainly discuss the convergence behavior of diffuse domain method (DDM) for solving semilinear parabolic equations with Neumann boundary condition defined in general irregular domains. We use a phasefield function to approximate the irregular domain and when the interface thickness tends to zero, the phasefield function will converge to indicator function of the original domain. With this function, we can modify the problem to another one defined on a larger rectangular domain that contains the targer physical domain. Based on the weighted Sobolev spaces, we prove that when the interface thickness parameter goes to zero, the numerical solution will converge to the exact solution. Also, we derive the corresponding optimal error estimates under the weighted L2 and H1 norms. Some numerical experiments are also carried out to validate the theoretical results.
Paper Structure (13 sections, 10 theorems, 73 equations, 4 figures, 3 tables)

This paper contains 13 sections, 10 theorems, 73 equations, 4 figures, 3 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. Allen-Cahn equation with double-well potential function
  13. Conclusions

Key Result

Theorem 3.1

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 (4)

  • Figure 1: Sketch of an example geometry: $D \subset D_{\epsilon} \subset \Omega$ 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 $\epsilon=1/64,1/32,1/16$ (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 results at the terminal time $T=0.5$ produced by the DDM approach with interface thickness $\epsilon=1/64,1/32,1/16$ (from left to right) for Example \ref{['ex2']} in the circular domain (top row) and the flower-shaped domain (bottom row).
  • Figure 4: The phase structures of the numerical errors at the terminal time $T$ produced by the DDM approach with interface thickness $\epsilon=1/32,1/16,1/8$ (from left to right) for Example \ref{['ex3']} in the flower-shaped domain.

Theorems & Definitions (16)

  • Theorem 3.1: Trace Theorem
  • Theorem 3.2: Embedding Theorem
  • Theorem 3.3: Poincare-Friedrichs-type inequality
  • Theorem 3.4
  • Theorem 3.5
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 4.1: Error estimate in the $L^2$ norm
  • proof
  • ...and 6 more