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

TL;DR

This paper establishes optimal error estimates for the diffuse domain method applied to semilinear parabolic equations with Neumann boundary conditions, demonstrating convergence as interface thickness diminishes.

Contribution

It provides the first rigorous convergence analysis and optimal error estimates for DDM applied to irregular domains in semilinear parabolic problems.

Findings

Convergence of the numerical solution to the exact solution as interface thickness tends to zero.

Derivation of optimal error estimates in weighted L2 and H1 norms.

Numerical experiments confirming theoretical convergence rates.

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.