Scholarlink

<p>In this paper, we introduce and investigate a novel numerical method for solving the Cahn–Hilliard equation in two-dimensional complex domains by employing region transformation. Initially, we convert the fourth-order equation into a second-order coupled system and formulate its first- and second-order semi-implicit schemes. Afterwards, we transform them into the polar coordinates equivalents. By introducing a category of weighted Sobolev spaces, we elaborate on fully discrete schemes and offer a theoretical validation of their stability. In particular, the introduction of pole singularities and the nonlinearity of the coupling problem pose significant challenges to theoretical analysis. To address these challenges, we introduce a novel class of projection operators and establish their approximation properties. Leveraging these properties, we provide error estimates for the approximate solutions. To validate our theoretical insights and algorithm’s efficacy, we conclude with a series of numerical examples.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Stability and error estimation based on a difference-spectral approximation for the Cahn–Hilliard equation in complex domains

  • 期刊论文

  • 发表日期：

    2025年3月15日

查看全文
  • Jihui Zheng,
  • Jing An

摘要

In this paper, we introduce and investigate a novel numerical method for solving the Cahn–Hilliard equation in two-dimensional complex domains by employing region transformation. Initially, we convert the fourth-order equation into a second-order coupled system and formulate its first- and second-order semi-implicit schemes. Afterwards, we transform them into the polar coordinates equivalents. By introducing a category of weighted Sobolev spaces, we elaborate on fully discrete schemes and offer a theoretical validation of their stability. In particular, the introduction of pole singularities and the nonlinearity of the coupling problem pose significant challenges to theoretical analysis. To address these challenges, we introduce a novel class of projection operators and establish their approximation properties. Leveraging these properties, we provide error estimates for the approximate solutions. To validate our theoretical insights and algorithm’s efficacy, we conclude with a series of numerical examples.