In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete $H^{-1}$ norm for the error function to establish the convergence result. In addition, the energy stability obtained in [Du et al., J. Comput. Phys., 363:39--54, 2018] requires an assumption on the uniform $\ell^\infty$ bound of the numerical solution and such a bound is figured out in this paper by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is $\ell^\infty$ bounded uniformly under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution.

中文翻译：

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非局部 Cahn-Hilliard 方程稳定线性半隐式数值格式的收敛分析

在本文中，我们为非局部 Cahn-Hilliard 方程的一阶稳定线性半隐式数值方案提供了详细的收敛分析，该方案遵循数值误差函数的一致性和稳定性估计。由于非线性项的形式复杂，我们对误差函数采用离散的$H^{-1}$范数来建立收敛结果。此外，[Du et al., J. Comput. Phys., 363:39--54, 2018] 需要对数值解的统一 $\ell^\infty$ 边界进行假设，本文通过进行高阶一致性分析来计算出这样的边界。考虑到数值解确实是有扰动的精确解，