Numerical treatment of singularities and the Generalized Finite Element Method: theory, algorithms, and applications

Numerical Methods for Partial Differential Equations (PDEs) are an

essential ingredient in many areas of Sciences and Engineering. Often

in the applications of PDEs, one has to deal with additional

difficulties caused by the singularities in the geometry, the

coefficients, or the boundary conditions. The main goals of this

proposal are to obtain more efficient numerical methods for equations

with singular solutions and to develop user friendly implementations.

The PI and his collaborators will design improved meshes that provide

optimal rates of convergence for the Finite Element Method in three

dimensions for elliptic equations on polyhedral domains with

non-smooth interfaces. These equations form one of the basic

ingredients in many other applications. The results will be extended

to quasi-linear elliptic equations and to evolution equations. For

some of these equations, spaces with higher smoothness or other additional

properties are sometimes needed, and the Generalized Finite Element

Method will be used for these purposes.

The proposal will contribute to the formation of graduate and

undergraduate students by advising and mentoring them and by

integrating them in research projects. The PI also plans to continue

to run an Interdisciplinary Seminar featuring outside speakers,

including non-mathematicians. The proposed research will lead to the

design and testing of new and improved numerical methods for Partial

Differential Equations (PDEs) of interest in Sciences and

Engineering. The resulting numerical methods will be presented and

implemented in a way that makes them accessible to

non-mathematicians. It will also contribute to applying mathematical

results in practice by interaction with researchers from other fields,

including the private sector. It will also contribute to the creation

of specialists able to use advanced theoretical tools to handle

practical problems. He will also introduce numerical methods and the

use of computers in more traditional undergraduate courses. The PI

also plans to popularize mathematical questions that arise from

practice among mathematicians. The theoretical results and the

resulting numerical methods will have applications in Mathematical

Physics, Quantum Chemistry, Biology, Finance (Risk Management and

pricing of Financial Derivatives, or Options), and other areas.