Research Experience in Numerical Methods for Partial Differential Equations with Singularities

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.