Discrete Maximal Parabolic Regularity for Time Discontinuous Galerkin Methods with Applications

时间不连续伽辽金方法的离散最大抛物线正则及其应用

基本信息

批准号：
1913133
负责人：
Dmitriy Leykekhman
金额：
$ 17.5万
依托单位：
University of Connecticut
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：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913133&HistoricalAwards=false
关键词：
Discrete Maximal Parabolic Regularity Time

项目摘要

Parabolic problems touch many areas of pure and applied mathematics and serve as a model of many environmental and real-life problems, such as optimal location of wastewater outfalls, location of the pollution sources, modeling of calcium waves in a heart cell and etc. Usually the resulting equations are very complicated to be treated analytically and must be solved by numerical methods. The analysis of such approximations is usually hard and technical and requires expertise in time and space discretization methods. For continuous problems the importance of the maximal parabolic regularity is well-recognized and has a number of applications, for example to nonlinear problems, optimal control problems and generally to problems where sharp results are required. In contrast to the continuous case, the discrete maximal parabolic regularity only recently came to the attention of the numerical analysis community and its potential is not yet fully realized. Such results, for example, reduce the analysis of transient problems to stationery ones, which are usually much less technical with many results already available in the literature. This for example would benefit a number of researchers, especially who are at the beginning of their careers and are not experts in time discretization methods.In this research the principal investigator will extend the theory of discrete maximal parabolic regularity for discontinuous Galerkin time schemes in several directions. The research plan includes the following projects. The first project extends the known results to non-symmetric autonomous elliptic operators, such as transient advection-reaction-diffusion problems, including the advection-dominated case. Such problems are classical and have been at the center of research for many years. As an application of such results, the PI intends to answer some of the open questions, for example whether the stabilized methods require stabilization parameters that depend on the time steps. In the second project, the PI intends to extend our previous results for non-autonomous problems to more general norms, that are important for a number of applications, for instance, quasilinear parabolic equations and optimal control problems. Finally, the PI plans to investigate more general parabolic systems, such as transient Stokes and Navier-Stokes problems, which are very important for the fluid flow problems and to obtain fully discrete best approximation type result in general Lebesgue space norms.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.

抛物线问题涉及纯数学和应用数学的许多领域，并可作为许多环境和现实生活问题的模型，例如废水排放口的最佳位置、污染源的位置、心脏细胞中钙波的建模等。由此产生的方程非常复杂，无法进行解析处理，必须通过数值方法求解。这种近似值的分析通常是困难且技术性的，需要时间和空间离散方法方面的专业知识。对于连续问题，最大抛物线正则性的重要性已得到广泛认可，并且具有多种应用，例如非线性问题、最优控制问题以及通常需要清晰结果的问题。与连续情况相反，离散最大抛物线正则性最近才引起数值分析界的注意，其潜力尚未完全实现。例如，这样的结果将瞬态问题的分析简化为固定问题，这些问题通常技术含量较低，许多结果已在文献中提供。例如，这将使许多研究人员受益，特别是那些刚刚开始职业生涯且不是时间离散方法专家的研究人员。在这项研究中，主要研究者将在几个方面扩展不连续伽辽金时间方案的离散最大抛物线正则性理论。方向。研究计划包括以下项目。第一个项目将已知结果扩展到非对称自治椭圆算子，例如瞬态平流反应扩散问题，包括平流主导的情况。此类问题是经典问题，多年来一直是研究的中心。作为此类结果的应用，PI 打算回答一些悬而未决的问题，例如稳定方法是否需要依赖于时间步长的稳定参数。在第二个项目中，PI 打算将我们之前针对非自治问题的结果扩展到更一般的规范，这对于许多应用都很重要，例如拟线性抛物线方程和最优控制问题。最后，PI计划研究更一般的抛物线系统，例如瞬态斯托克斯和纳维-斯托克斯问题，这些问题对于流体流动问题非常重要，并在一般勒贝格空间范数中获得完全离散的最佳近似类型结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。