Approximation of Singular Solutions to Nonlocal and Nonlinear Models

数值分析在发展和分析理论和应用科学中的经典模型的近似解方面非常成功。然而，在最近的时代，新的模型类已经出现了挑战目前的理解和数值分析技术。需要新的近似技术，或者对经典技术的分析需要新的想法，因为标准论点不起作用。这是特别的情况下，表现出非局部功能的时间（记忆效应），非光滑演化问题，非局部功能的空间（长程相互作用），或任何这些功能的组合。另一类需要特别注意的重要问题是那些问题的数据是非光滑的，其中包括奇异强迫或本构律。最后，作为最后一个例子，有一个障碍，在如何顺利的解决方案是强非线性问题，无论问题的光滑度data.This研究项目的目的是分析的近似技术的一个代表性的样本上述问题。我们将讨论的许多数值技术在许多情况下的实现是标准的，但它们的分析需要解的正则性（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，新的数值技术将被开发，现有的将加强其近似性能的坚实的数学分析。我们研究的模型描述了广泛的现象，并将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，非局部性也大大复杂化了求解方案的分析和有效实施。

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.