Research Experience in Numerical Methods for Partial Differential Equations with Singularities

计算方法几乎用于任何使用计算机的领域。改进的数值和计算方法是提高现代计算机应用速度的主要途径之一。“有限元法”是一种非常流行的数值方法，被工程师、物理学家、生物学家、气象学家、金融数学和许多其他领域所使用。更具体地说，有限元方法被用于热传播、雷达探测、医学成像、建筑物结构分析、飞机机翼设计以及许多其他实际问题的研究中。使用有限元法的第一步是将待研究的区域（建筑物、飞机、待成像物体所占据的区域）虚拟地划分为许多小三角形、平行六面体或四面体。这在很大程度上是一项基本任务，但非常耗时。此外，在靠近域的顶点和边缘的地方需要额外的注意。如果在划分定义域的方式上没有这种额外的注意，获得所需的计算精度将花费更长的时间。例如，最近一些主要研究者也参与的结果导致这样的估计：对于三个精确数字的精度，一个仔细的除法程序可以将计算时间减少一百万倍或更多。对于五个精确数字的精度，估计的改进是一万亿倍或更多。虽然决定计算域的改进划分的形状的数学技术是相当复杂的（各向异性网格细化，椭圆偏微分方程，非紧流形，泛函分析），一旦这样的划分算法已经制定，它可以教授给一个好的本科初学者。本提案的主要目的是培养本科生和研究生掌握最先进的有限元方法技术（包括但不限于各向异性网格细化、先验和自适应网格细化、无网格方法和求解所得线性系统的多网格方法）。学生越高级，他们就有越多的机会了解这些方法的内部工作原理。每个参与的学生都将被期望在他们相应的数学训练水平上做出原创性的研究。由此产生的学生培训将为学生、首席研究员和其他人提供必要的数字经验，以便在未来进行前沿研究。有限元法的另一个重要特点是，它可以教授给具有非常广泛的数学背景和因此复杂的学生。因此，根据以往的经验，通过培训更多具有不同背景的人使用这些方法并对其进行研究，预计将吸引更广泛的学生，包括代表性不足的群体，从事数学研究。

Computational methods are used in almost any area in which computers are used. Improved numerical and computational methods are one of the main venues to improve the speed of modern computer applications. The "Finite Element Method" is a very popular numerical method that is used by engineers, physicists, biologists, meteorologists, in financial mathematics, and in many other areas. More concretely, the finite element method is used in the study of heat propagation, radar detection, medical imaging, structural analysis of buildings, design of aircraft wings, and in many other practical problems.The first step in the use of the Finite Element Method is to fictitiously divide the domain to be studied (the domain occupied by a building, aircraft, body to be imaged) into many small triangles, parallelepipeds, or tetrahedra. This is to a large extent an elementary task, but very time-consuming. Moreover, additional care has to be paid close to the vertices and the edges of the domain. Without this additional extra care in the way we divide our domain, obtaining the desired precision in calculations would take much longer. For example, some recent results in which the Principal Investigator is also involved lead to the estimate that, for a precision of three exact digits, a careful division procedure can decrease the time of calculation by one million times or more. For a precision of five exact digits, the estimated improvement is one trillion times or more.While the mathematical techniques to decide the shape of the improved divisions of the computational domain are quite sophisticated (anisotropic mesh refinement, elliptic partial differential equations, non-compact manifolds, functional analysis), once such a division algorithm has been formulated, it can be taught to a good beginning undergraduate student. The main purpose of this proposal is to train undergraduate and graduate students in state-of-the-art Finite Element Method techniques (including, but not limited to, anisotropic mesh refinement, a priori and adaptive mesh refinement, meshless methods, and multigrid methods for solving the resulting linear systems). The more advanced the students, the more opportunities they will be given to learn about the inner workings of these methods. Each of the students involved will be expected to produce original research at their corresponding level of mathematical training. The resulting training of the students will provide the necessary numerical experience needed by the students, by the Principal Investigator, and by others to conduct cutting-edge research in the future. Another important quality of the Finite Element Method is that it can be taught to students with a very wide scale of mathematical backgrounds and hence sophistication. As such, by training more people with various backgrounds to use such methods and to do research on them, it is expected, based on previous experience, that a wider range of students, including underrepresented groups, will be attracted to mathematical research.