In this work, we consider parabolic models with dynamic boundary conditions and parabolic bulk-surface problems in 3D. Such partial differential equations–based models describe phenomena that happen both on the surface and in the bulk/domain. These problems may appear in many applications, ranging from cell dynamics in biology, to grain growth models in polycrystalline materials. Using difference potentials framework, we develop novel numerical algorithms for the approximation of the problems. The constructed algorithms efficiently and accurately handle the coupling of the models in the bulk and on the surface, approximate 3D irregular geometry in the bulk by the use of only Cartesian meshes, employ fast Poisson solvers, and utilize spectral approximation on the surface. Several numerical tests are given to illustrate the robustness of the developed numerical algorithms.

中文翻译：

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具有动态边界条件和体面问题的模型的差分势方法

在这项工作中，我们考虑具有动态边界条件的抛物线模型和3D中的抛物线体表面问题。这种基于偏微分方程的模型描述了在表面以及在体/域中都发生的现象。这些问题可能出现在许多应用中，从生物学中的细胞动力学到多晶材料中的晶粒生长模型。使用差势框架，我们开发了新颖的数值算法来解决问题。所构造的算法可以高效，准确地处理模型在主体和表面上的耦合，仅使用笛卡尔网格即可在主体中近似3D不规则几何，采用快速泊松解算器，并利用表面上的光谱近似。