Approximation of Singular Solutions to Nonlocal and Nonlinear Models

数值分析在理论科学和应用科学中近似经典模型解的格式的开发和分析方面已经非常成功。然而，最近出现了一些新的模型，它们对现有的数值分析的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，因为标准的论点不起作用。对于在时间上表现出非局部特征(记忆效应)、非光滑演化问题、在空间上表现出非局部特征(长距离交互作用)或这些特征的任何组合的问题，情况尤其如此。另一类需要特别注意的重要问题是问题的数据是非光滑的，其中包括奇异强迫或本构定律。最后，作为最后一个例子，在强非线性问题中，无论问题数据的光滑性如何，在解的光滑程度上都存在障碍。本研究项目的目的是分析上述问题中具有代表性的样本的逼近技术。我们将在许多情况下讨论的许多数值技术的实现是标准的，但它们的分析要求(在非标准空间中)解的正则性、问题的结构和格式的结构之间的良好相互作用。作为这项工作的结果，将开发新的数值技术，并通过对其逼近性质的坚实的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，并将开发出针对这些现象的数学上可靠的数值方法。因此，提出的想法将增强建模和预测能力。例如，对非局部算子的离散化技术的研究还处于起步阶段。即使在线性情况下，非局部性也使解决方案的分析和有效实施变得非常复杂。

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.