Solving two and three dimensional Interface Problems using Non-body-fitted Grids

一个典型的椭圆界面上的问题被转化为不同区域的分段定义的椭圆型偏微分方程组(PDE)，这些偏微分方程组与界面条件(如解的跳跃和界面上的通量)相耦合。在许多情况下，例如当界面移动时，挑战是如何在不生成贴体网格的情况下准确、稳健和高效地解决这样的问题。关键问题是如何在固定网格上有效地捕捉界面的复杂几何形状和界面上的跳跃条件，而界面不与网格对齐，且PDE在界面上无效。PI提出了一种非传统的有限元方法来求解椭圆和弹性界面问题。以下是提出的项目：1.解决三维椭圆界面问题；2.解决三维弹性界面问题；3.解决三维多域界面问题；4.解决二维多域弹性界面问题；5.解决运动界面问题；6.跨学科应用。界面问题在电磁学、材料科学、流体力学等领域有着广泛的应用。例如，界面可以是蛋白质的分子表面、水-油界面或类似的界面。因此，通过改进用于模拟界面的数值技术，拟议的研究将对跨学科应用产生重大影响。此外，尽管研究需要先进的技术，但界面的概念存在于日常生活中，这意味着可以在应用数学研讨会中激发学生的兴趣。

A typical elliptic interface problem is cast as piecewise defined elliptic partial differential equations (PDE) in different regions which are coupled with interface conditions, such as jumps in the solution and in the flux across the interface. In many situations, such as when the interface is moving, the challenge is how to solve such a problem accurately, robustly and efficiently without generating a body fitted mesh. The key issue is how to capture the complex geometry of the interface and the jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the PDE is not valid across the interface. The PI proposes a non-traditional finite element method to solve the elliptic and elasticity interface problems. The following are the proposed projects: 1. solving 3D elliptic interface problems; 2. solving 3D elasticity interface problems; 3. solving 3D multi-domain interface problems; 4. solving 2D multi-domain elasticity interface problems; 5. solving moving interface problems; 6. interdisciplinary applications. Interface problems have applications in electromagnetics, material science, fluid dynamics and other areas. For example, the interface could be the molecular surface of a protein, the water-oil interface, or similar. Therefore, by improving numerical techniques used to model interfaces, the proposed research will have substantial impact on interdisciplinary applications. Further, although the research requires advanced techniques, the concept of an interface exists in everyday life, which means that student interest can be stimulated in applied mathematics seminars.