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

非对齐网格界面问题的高阶方法

• 批准号: 1522663
• 负责人: Marcus Sarkis
• 金额: $ 18.93万
• 依托单位: Worcester Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522663&HistoricalAwards=false
• 关键词: Higher; Order; Methods; Interface; Problems

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.

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，多孔介质流中的前沿跟踪和材料科学，仅举几例。这些问题的困难之一是，解决方案通常是不光滑的界面，因此，标准的数值方法将失去界面附近的精度，除非网格对齐。然而，它是有利的，有网格不对齐的界面，特别是对于时间依赖的问题，界面随时间移动。在每一个时间步重新划分网格可能代价高昂，可能破坏网格的结构，可能恶化刚度矩阵的良好条件，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度有限元浸没边界法（FE-IBM）的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的界面问题的鲁棒高阶离散的设计和分析。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、多孔介质流动、材料科学和并行计算。该项目将影响流体-结构相互作用社区中使用的数值算法的发展。一个更广泛的影响将是培养研究生和本科生的数学和相关学科，让他们接触到跨学科的问题和合作解决问题的重大技术importance.One的有限元和有限差分浸入边界方法的缺点是，他们只有一阶精度，由于非光滑的解决方案在整个接口。此外，这些方法存在的时间相关的问题和流体-结构相互作用问题的数学分析很少。该项目的第一部分涉及高阶FE-IBM算法的建设，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。该项目的第二部分涉及新的有限元方法，能够准确地捕捉高对比度系数的情况下，有限元网格不一定与界面对齐的椭圆界面问题的解决方案。这里的目标是开发具有最佳收敛速度的有限元方法，其中隐藏在这些估计中的常数与对比度和网格如何穿过界面无关。