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.