Approximation of Singular Solutions to Nonlocal and Nonlinear Models

非局部和非线性模型奇异解的逼近

基本信息

批准号：
1720213
负责人：
Abner Salgado
金额：
$ 16.67万
依托单位：
University of Tennessee Knoxville
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720213&HistoricalAwards=false
关键词：
Approximation Singular Solutions Nonlocal Nonlinear

项目摘要

Numerical analysis has been very successful in the development and analysis of schemes to approximate the solution of classical models in the pure and applied sciences. However, in recent times, new classes of models have emerged which challenge the current understanding and techniques of numerical analysis. New approximation techniques are required or the analysis of the classical ones call for new ideas, as standard arguments do not work. This is particularly the case for problems which exhibit nonlocal features in time (memory effects), nonsmooth evolution problems, nonlocality features in space (long range interactions), or a combination of any of these features. Another important class of problems that require special attention are those where the data of the problem is nonsmooth, which includes singular forcing or constitutive laws. Finally, as a last example there are strongly nonlinear problems where there is a barrier in how smooth the solution can be, regardless of the smoothness of the problem data.The purpose of this research project is the analysis of approximation techniques for a representative sample of the problems mentioned above. The implementation of many of the numerical techniques that we will discuss in many cases is standard, but their analysis requires a fine interplay between the regularity of the solution (in nonstandard spaces), the structure of the problem and that of the scheme. As an outcome of this work, new numerical techniques will be developed and the existing ones will be strengthened by solid mathematical analysis of their approximation properties. The models which will be under our study describe a wide range of phenomena, and mathematically solid numerical methods for them will be developed. Thus, the proposed ideas will enhance modeling and prediction capabilities. For instance, the study of discretization techniques for nonlocal operators is in its infancy. Even in the linear case, the nonlocality greatly complicates the analysis and efficient implementation of solution schemes.

数值分析在纯科学和应用科学中近似经典模型解的方案的开发和分析方面非常成功。然而，近年来，新型模型的出现对当前的数值分析理解和技术提出了挑战。需要新的近似技术，或者对经典技术的分析需要新的想法，因为标准论证不起作用。对于表现出时间非局部特征（记忆效应）、非平滑演化问题、空间非局部特征（长程相互作用）或任何这些特征的组合的问题尤其如此。另一类需要特别注意的重要问题是问题数据不光滑的问题，其中包括奇异强迫或本构定律。最后，作为最后一个例子，存在强非线性问题，无论问题数据的平滑程度如何，解决方案的平滑程度都存在障碍。本研究项目的目的是分析具有代表性的样本的近似技术上述问题。我们将在许多情况下讨论的许多数值技术的实现都是标准的，但它们的分析需要解决方案的规律性（在非标准空间中）、问题的结构和方案的结构之间的良好相互作用。作为这项工作的成果，将开发新的数值技术，并通过对其近似性质的可靠数学分析来加强现有的数值技术。我们将研究的模型描述了广泛的现象，并将为它们开发数学上可靠的数值方法。因此，所提出的想法将增强建模和预测能力。例如，非局部算子离散化技术的研究还处于起步阶段。即使在线性情况下，非局部性也会使解决方案的分析和有效实现变得非常复杂。