In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn–Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn–Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L ∞ L^{\infty} -norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.

中文翻译：

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具有浓度相关迁移率的 Cahn-Hilliard 方程的无条件能量稳定局部不连续伽辽金格式的误差分析

在本文中，我们主要研究具有浓度依赖迁移率的 Cahn-Hilliard 方程的无条件能量稳定局部不连续伽辽金 (LDG) 方案的误差分析。时间离散化基于不变能量二次方 (IEQ) 方法。完全离散的方案导致线性代数系统在每个时间步上求解。误差估计的主要困难是对 LDG 离散化中单元边界处的某些跳跃项缺乏控制。需要对 Cahn-Hilliard 方程的初始条件和非恒定迁移率项进行特殊处理。对于非常数迁移项的分析，我们充分利用半隐式时间离散方法，通过数学归纳法将一些数值变量约束在L ∞ L^{\infty} -范数上。