Novel Virtual Element Methods with Applications in Interface Problems

界面问题产生于许多重要的复杂多物理场和生物系统，如涉及多流体/材料界面的演变，肿瘤生长或干细胞变形。计算机辅助模拟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。虚拟单元法是科学计算领域中一种新兴的强有力的工具，可以用来逼近这些系统的控制方程。本研究项目的目标是发展各种VEM的理论和实践方面。能够可靠地回答“我们可以信任我们的模拟结果吗？证明了在模拟这些复杂系统时使用VEM的合理性。同时，该项目致力于为公众提供最先进的VEM计算机程序，以节省宝贵的计算资源。此外，本研究项目还为研究生创造了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。本计画将针对椭圆型界面问题，发展一个深入的稳健的VEM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收敛对多面体网格几何形状的依赖性，证明了VEM对界面拟合网格的适用性，这些网格可能在界面附近变得非常不规则或退化。同时，本项目借鉴VEM框架的新奇，改进了传统接口问题分析方法。VEM的椭圆问题的元公式使我们能够自然地在高阶和/或3-D中构造浸入式有限元空间。 高阶界面VEM，后验误差估计，和自适应多面体网格细化的研究，使VEM更高效和有效的。这种综合研究使我们能够攻击具有挑战性的3-D接口问题，这反过来又拓宽了整个数值偏微分方程社区的理论和工具的范围。最后但并非最不重要的是，一个可移植的和高度矢量化的VEM软件库应公开提供，包括半结构化的界面拟合网格生成，矢量化组装，多面体自适应，和快速多重网格求解器。计算机代码的可移植性使研究人员能够将VEM纳入现有的软件库中处理接口问题，从而促进跨学科的研究，以模拟这些复杂的系统。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.