Novel Virtual Element Methods with Applications in Interface Problems

新颖的虚拟元素方法及其在界面问题中的应用

• 批准号: 1913080
• 负责人: Shuhao Cao
• 金额: $ 15.36万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913080&HistoricalAwards=false
• 关键词: Novel; Virtual; Element; Methods; Applications

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.

界面问题来自许多重要的复杂多物理学和生物系统，例如涉及多流体/材料界面，肿瘤生长或干细胞变形的界面问题。计算机辅助模拟是针对这些具有挑战性的界面问题进行研究的经济友好工具。为了近似这些系统的数学方程，虚拟元素方法（VEM）是科学计算社区中新兴的强大工具。该研究项目的目的是开发各种VEM的理论和实际方面。能够可靠地回答“我们可以相信我们的模拟结果吗？”证明使用VEM在模拟这些复杂系统中的使用是合理的。同时，该项目努力为公众提供最先进的VEM计算机程序，以节省有价值的计算资源。此外，该研究项目创造了将火炬传递给研究生的机会，成为下一代计算数学家。使用高对比度扩散系数来解决椭圆形偏微分方程，在建模这些复杂系统中起着核心作用。该项目应在椭圆界面问题上对VEM进行深入的鲁棒分析。与现有的VEM分析不同，该项目设计了一个新的新型范式来研究界面VEM的误差分析，并进一步阐明了VEM收敛对多面网格几何形状的依赖性，证明VEM对接口拟合的网格的适用性是合理的。同时，该项目从VEM框架的新颖性中学习，以改善对界面问题的传统方法的分析。 VEM针对椭圆问题的元元素使我们能够自然地以更高的顺序和/或3-D构建浸泡的有限元空间。 高阶界面VEM，A后验误差估计和自适应多层网格的细化，以使VEM更有效，有效。这项综合研究使我们能够攻击具有挑战性的3-D接口问题，这反过来又扩大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一点是，应公开提供便携式且高度矢量化的VEM软件库，包括半结构化接口拟合的网格生成，矢量化组装，多层适应性和快速的Multigrid solvers。计算机代码的可移植性使研究人员能够将VEM纳入处理界面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NSF的法定任务，并认为通过使用该基金会的知识分子和更广泛的影响来评估Criteria的评估，并被视为值得通过评估的支持。

项目成果

Finite Elements for div- and divdiv-Conforming Symmetric Tensors in Arbitrary Dimension

• DOI: 10.1137/21m1433708
• 发表时间: 2021-06
• 期刊: SIAM J. Numer. Anal.
• 作者: Long Chen;Xuehai Huang

Transformed primal-dual methods for nonlinear saddle point systems

非线性鞍点系统的变换原对偶方法

• DOI: 10.1515/jnma-2022-0056
• 发表时间: 2023
• 期刊: Journal of Numerical Mathematics
• 影响因子: 3
• 作者: Chen, Long;Wei, Jingrong
• 通讯作者: Wei, Jingrong

Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions

• DOI: 10.1142/s0218202523500112
• 发表时间: 2022-02
• 期刊: Mathematical Models and Methods in Applied Sciences
• 影响因子: 3.5
• 作者: Shuhao Cao;Long Chen;Ruchi Guo

A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions

• DOI: 10.1016/j.camwa.2022.03.015
• 发表时间: 2021-01
• 期刊: Comput. Math. Appl.
• 作者: Shuhao Cao;Chunmei Wang;Junping Wang

Anisotropic Error Estimates of the Linear Nonconforming Virtual Element Methods

• DOI: 10.1137/18m1196455
• 发表时间: 2018-06
• 期刊: SIAM J. Numer. Anal.
• 作者: Shuhao Cao;Long Chen