Models based on partial differential equations are widely used in science and engineering, and numerical methods for their solutions are of central interest. In the last several years, there has been steadily growing interest in two such methods: Generalized Finite Element Methods and the closely related Meshless Methods. They have become a focus of development and application -- especially in the engineering community. These methods generalize the Finite Element Method, and are of interest mainly for two reasons: (i) they are not based on a mesh, or use a mesh only minimally, and hence avoid some of the problems in meshing complicated domains; (ii) they permit the use of non-polynomial approximating functions, and hence are applicable to problems with special features, whose solutions are not accurately approximated by polynomials. This flexibility has, in the past few years, been effectively used in handling certain important problems. But a deep and precise theoretical understanding of these methods is needed to broaden their applicability, in particular their efficiency and robustness, to complex problems in engineering that otherwise cannot be solved with existing methods, for example, the Finite Element Method.
This project involves careful and precise mathematical study of various aspects of Generalized Finite Element and Meshless Methods. The broader impact of this study will be reflected in the applicability of these methods to a wide range of complex problems of technical and societal importance; for example, problems with multiple scales. Multiscale problems are increasingly important; they include the problems of composite materials and flow through porous media. Finding robust, efficient, and reliable approaches to solve these problems is one of the challenges of modern technology. Another impact of the proposal is the training of graduate students in a vital, cutting edge area. Their involvement in this project will not only help them become experts in the mathematical underpinning of these important methods, but will also give them the rare first-hand experience in working with practical engineering problems.
