We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the re-formulated equations converge to those of the original equations. We provide numerical simulations which confirm this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy.

中文翻译：

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求解复杂几何中的 pdes

我们扩展了以前的工作，并提出了一种使用 Dirichlet、Neumann 和 Robin 边界条件求解复杂、静止或移动几何中的偏微分方程的通用方法。通过辅助相场函数使用几何的隐式表示，用扩散层（例如，扩散域）代替域的尖锐边界，在更大的规则域上重新公式化方程。生成的偏微分方程与原始方程的阶数相同，并具有额外的低阶项来逼近边界条件。重新公式化的方程可以通过标准数值技术求解。我们使用匹配渐近展开的方法来证明重新制定方程的解收敛于原始方程的解。我们提供的数值模拟证实了这一分析。我们还介绍了该方法在生长域和复杂的三维结构中的应用，并讨论了在细胞生物学和异质外延中的应用。