In its broad outlines, the research program of the P.I. aims at the development, analysis and computer implementation of numerical methods designed to approximate the solutions of some partial differential equations (pde's) that have important applications in the fields of engineering and physics. The Discontinuous Galerkin method constitutes the core methodology of this effort. The ultimate goal of the research is to make full use of this method to develop convergent and efficient adaptive methods designed to reduce the run time of the algorithms by finding optimal or quasi-optimal meshes. Other efforts will be directed towards the development of domain decomposition and multigrid algorithm for the fast solution of the resulting systems of equations on single as well as multiprocessor computers. The P.I. will develop methods and adaptive algorithms for second and fourth order elliptic problems, the incompressible Navier-Stokes equations and the Cahn-Hilliard equations and use them to simulate phenomena modeled by these equations. Adaptive methods will also be used to simulate finite-time blowup of nonlinear evolution equations. Some areas of applications are chemical reactions where the geometry of the domain plays an important role and the simulation of tumor growth.
Scientific computing is playing an increasingly important role in the progress of Science as a cost effective alternative to "real life" experiments which could be very costly, say wind tunnel experiments in aircraft design, or even impossible to duplicate, such as supernova explosions and other astrophysical phenomena. It is worth mention that numerical simulations are playing a crucial role in the identification of the potential effects of global warming. The U.S. government, through its various funding agencies, has made important investments by the creation of supercomputing centers equipped with the latest generation of massively parallel computers. To extract the full power of these machines, with some having tens of thousands of individual processors, efficient and "scalable" methods and algorithms must be developed to keep pace with the advances in hardware. Indeed, most current algorithms fall short of harnessing the full power of these computers especially when the number of processors exceeds a few thousand. The Discontinuous Galerkin method is a recently introduced methodology with great potential and wide applicability. Its many attributes include flexibility, ability to handle complex geometries and scalability. Further understanding of this approach and the development of efficient and parallel computer codes will have a positive impact on the ever increasing areas of Science that make essential use of numerical simulations. Finally, two graduate students are actively participating in this project as partial fulfillment of their Ph.D. degree requirements. This will achieve another goal of this project which is to contribute to the training of the next generation of researchers.
