Multiscale Methods for Partial Differential Equations

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DMS Award AbstractAward #: 0209497PI: Xu, JinchaoInstitution: Pennsylvania State University Program: Computational MathematicsProgram Manager: Catherine MavriplisTitle: Multiscale Methods for Partial Differential EquationsThe focus of this work is on the development and applications of a two-scale discretization technique, namely the finite element method based on partition of unity. One main application is on the design of efficient discretization for nonmatching (either overlapping or nonoverlapping) grids. The main idea of nonmatching grids is to divide a physical domain into a set of overlapping or nonoverlapping subregions which can accommodate smooth, simple, easily generated grids. In this approach, a grid generation for complex geometries can be made simple, refinement grids can be added or removed without changing other grids, different equations/numerical methods may be used on different grids, efficient structured grid solvers may be used. Furthermore, overlapping grids are well suited for parallelization and vectorization. The proposed generalized finite element method based on partition of unity provides a general and powerful discretization framework for this type of grids. Another major task is the development of a multigrid iterative method for solving the resulting algebraic systems for these new discretization schemes. As divide and conquer techniques, the proposed multiscale algorithms are suitable for parallel and high-performance computers. A class of new multiscale techniques are proposed to study for efficient numerical solution of partial differential equations. Multiscale methods in general are proven to be among the most powerful mathematical tools for the investigation of a broad range of models that are described by partial differential equations. Their pivotal role in the design of fast, reliable, and robust numerical methods for the solution of various problems places them among the most important research areas in the applied mathematics in the recent years. Since these methods are in some sense problem-independent, they are expected to have many important applications in science and engineering such as composite materials and subsurface flows in environmental applications.Date: May 28, 2002