Novel Virtual Element Methods with Applications in Interface Problems

界面问题出现在许多重要的复杂的多物理场和生物系统中，例如涉及多流体/材料界面的进化、肿瘤生长或干细胞变形的系统。计算机辅助仿真是研究这些具有挑战性的界面问题的一种成本友好的工具。为了逼近这些系统的控制数学方程，虚元法（VEM）是科学计算界新兴的强大工具。本研究项目的目标是发展各种虚拟机械设备的理论和实践方面。能够可靠地回答“我们可以信任我们的仿真结果吗？”这个问题，证明了在模拟这些复杂系统时使用VEMs是合理的。同时，本项目力求为公众提供最先进的VEM计算机程序，从而节省宝贵的计算资源。此外，这个研究项目创造了将火炬传递给研究生成为下一代计算数学家的机会。求解具有高对比度扩散系数的椭圆型偏微分方程在这些复杂系统的建模中起着核心作用。本课题将对椭圆界面问题的向量机进行深入的鲁棒先验误差分析。与现有的VEM分析不同，本项目设计了一种新的范式来研究界面VEM的误差分析，并进一步阐明了VEM收敛对多边形网格几何形状的依赖，证明了VEM适用于在界面附近可能变得极其不规则或退化的界面拟合网格。同时，本项目借鉴了VEM框架的新颖性，改进了传统方法对接口问题的分析。VEM的椭圆问题元公式使我们能够自然地构建高阶和/或三维的浸入式有限元空间。研究了高阶界面向量机、后验误差估计和自适应多边形网格细化等方法，提高了向量机的效率和有效性。这项综合研究使我们能够解决具有挑战性的三维界面问题，这反过来又拓宽了整个数值偏微分方程社区在理论和工具方面的范围。最后但并非最不重要的是，应公开可移植和高度矢量化的VEM软件库，包括半结构化接口拟合网格生成，矢量化组装，多面自适应和快速多网格求解器。计算机代码的可移植性使研究人员能够将VEM整合到处理接口问题的现有软件库中，从而促进了模拟这些复杂系统的跨学科研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Interface problems arise from many important complex multiphysics and biological systems, such as those involving the evolutions of multi-fluid/material interfaces, tumor growth, or stem cell deformation. Computer-aided simulation is a cost-friendly tool for the studies of these challenging interface problems. To approximate the governing mathematical equations of these systems, Virtual element method (VEM) is an emerging powerful tool in the scientific computing community. The objective of this research project is to develop both theoretical and practical aspects of various VEMs. Being able to reliably answer the question "can we trust our simulation results?" justifies the use of VEMs in simulating these complex systems. Meanwhile, this project strives to provide the public with a state-of-the-art VEM computer program that saves valuable computing resources. In addition, this research project creates opportunities to pass the torch on to graduate students to become the next generation computational mathematicians.Solving elliptic partial differential equations with high-contrast diffusion coefficients play a central role in the modeling these complex systems. This project shall develop an in-depth robust a priori error analysis of VEM on elliptic interface problems. Different from the existing VEM analysis, this project devises a new novel paradigm to study the error analysis for the interface VEM, and further clarifies the dependence of the VEM convergence on the polytopal mesh geometries, justifying the VEM's applicability on interface-fitted mesh which may become extremely irregular or degenerate near the interfaces. Meanwhile, this project learns from the novelty of VEM framework to improve the analyses of traditional approaches for interface problems. The VEM's meta-formulation for elliptic problems enables us to construct immersed finite element spaces naturally in higher order and/or in 3-D. Higher order interface VEMs, the a posteriori error estimation, and the adaptive polytopal mesh refinement are to be studied to render VEM more efficient and effective. This integrated study enables us to attack the challenging 3-D interface problems, which, in turn, broadens the scope in terms of both theory and tools for the whole numerical partial differential equation community. Last but not least, a portable and highly-vectorized VEM software library shall be made publicly available, including the semi-structured interface-fitted mesh generation, vectorized assembling, polytopal adaptivity, and fast multigrid solvers. The portability of the computer code enables the researchers to incorporate the VEM into existing software libraries dealing with interface problems, thus facilitating the interdisciplinary research in simulating those complex systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.