Higher-Order Methods for Interface Problems with Non-Aligned Meshes

界面问题出现在几个应用中，包括心脏模型、耳蜗模型、水生动物运动、血细胞运动、多孔介质流动的前沿跟踪和材料科学，仅举几例。这些问题的困难之一是解决方案通常在界面上不光滑，因此标准数值方法将在界面附近失去精度，除非网格对准它。然而，拥有不与界面对齐的网格是有利的，特别是对于界面随时间移动的时间相关问题。在每一个时间步重新网格划分可能会非常昂贵，可能会破坏网格的结构，可能会恶化刚度矩阵的良好调理，并影响问题的稳定性。研究的第一个问题将涉及新的稳定和高阶精确的有限元浸入边界方法（FE-IBM），用于界面随时间移动的演化问题。研究的第二个问题是高对比度不连续扩散系数界面问题的鲁棒高阶离散化设计和分析。该项目的好处包括加强数值分析与其他科学和工程领域之间的联系，特别是生物工程、多孔介质流、材料科学和并行计算。这个项目将影响流固相互作用领域中使用的数值算法的发展。更广泛的影响将是培养数学及相关学科的研究生和本科生，让他们接触跨学科问题，并通过合作解决具有重大技术重要性的问题。有限元法和有限差分浸入边界法的缺点之一是，由于解在界面上的非光滑性，它们只能达到一阶精度。此外，对于时间相关问题和流固耦合问题，这些方法的数学分析很少。项目的第一部分涉及构建高阶FE-IBM算法并建立相应的数学基础，以获得严格的时间稳定性和先验和后验误差估计。项目的第二部分涉及新的有限元方法，该方法能够在有限元网格不一定与界面对齐的情况下准确捕获具有高对比度系数的椭圆界面问题的解。这里的目标是开发具有最佳收敛率的有限元方法，其中隐藏在这些估计中的常数与对比度无关，也与网格如何穿过界面无关。

Interface problems arise in several applications including heart models, cochlea models, aquatic animal locomotion, blood cell motion, front-tracking in porous media flows and material science, to name a few. One of the difficulties in these problems is that solutions are normally not smooth across interfaces, and therefore standard numerical methods will lose accuracy near the interface unless the meshes align to it. However, it is advantageous to have meshes that do not align with the interface, especially for time dependent problems where the interface moves with time. Re-meshing at every time step can be prohibitively costly, can destroy the structure of the grid, can deteriorate the well-conditioning of the stiffness matrix, and affect the stability of the problem. The first problem studied will involve new stable and higher-order accurate Finite Element - Immersed Boundary Methods (FE-IBM) for evolution problems where the interface moves with time. The second problem studied is the design and analysis of robust higher-order discretizations for interface problems with high-contrast discontinuous diffusion coefficients. Benefits of the project include the strengthening of connections between numerical analysis and other areas of science and engineering, particularly bioengineering, porous media flows, material sciences and parallel computing. This project will impact the development of numerical algorithms used in the fluid-structure interaction communities. A broader impact will be the training of graduate and undergraduate students of mathematics and related disciplines by exposing them to interdisciplinary problems and collaborations addressing questions of great technological importance.One of the drawbacks of the finite element and finite difference immersed boundary methods is that they are only first-order accurate due to the non-smoothness of the solution across the interface. Furthermore, very few mathematical analyses of these methods exist for time dependent problems and for fluid-structure interaction problems. The first part of the project involves the construction of higher-order FE-IBM algorithms and establishing a corresponding mathematical foundation to obtain rigorous time stability and a priori and a posteriori error estimates. The second part of the project deals with new finite element methods which are able to accurately capture solutions of elliptic interface problems with high-contrast coefficients in the case that the finite element mesh is not necessarily aligned with the interface. The goal here is to develop finite element methods with optimal convergence rates, where the constants hidden in these estimates are independent of the contrast and on how the mesh crosses the interface.