Approximation of Singular Solutions to Nonlocal and Nonlinear Models

在纯科学和应用科学中，数值分析已经非常成功地开发和分析了近似经典模型解的格式。然而，近年来出现了一些新的模型，它们对当前的数值分析的理解和技术提出了挑战。由于标准论证不起作用，需要新的近似技术，或者对经典近似的分析需要新的思想。这尤其适用于那些在时间上表现出非局部特征（记忆效应）、非光滑进化问题、空间上表现出非局部特征（远程相互作用）或这些特征的组合的问题。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，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对上述问题的代表性样本的近似技术进行分析。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，但它们的分析需要在解的规律性（在非标准空间中）、问题的结构和方案的结构之间进行良好的相互作用。作为这项工作的结果，新的数值技术将得到发展，现有的数值技术将通过对其近似性质的可靠数学分析得到加强。我们将研究的模型描述了广泛的现象，并且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，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，非定域性也会使求解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.