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.

翻译：本文主要探讨扩散域方法(DDM)在求解定义于一般不规则区域上、具有Neumann边界条件的半线性抛物方程时的收敛性。我们采用相场函数逼近不规则区域，当界面厚度趋于零时，该相场函数将收敛至原区域的指示函数。借助此函数，可将原问题转化为定义在包含目标物理域的更大矩形区域上的修正问题。基于加权Sobolev空间理论，我们证明了当界面厚度参数趋于零时，数值解将收敛于精确解。同时，我们在加权L2范数与H1范数下推导了相应的最优误差估计。通过数值实验验证了理论结果的正确性。