Adaptive Discontinuous Galerkin Methods and Applications

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In its broad outlines, the research program outlined in this proposal aims at the development, analysis and computer implementation of numerical methods designed to approximate the solutions of some partial differential equations with important applications in the fields of Engineering, Physics, the Biomedical Sciences and Mathematics. While the discontinuous Galerkin method will constitute the core methodology of this effort, a wide range of themes will be covered: A priori and a posteriori error estimates, adaptive methods, efficient solution of linear and nonlinear systems using multigrid and domain decomposition methods, flexible data structures and implementation on multicore and parallel processors and finally, applications to practical and analytical problems. A particularly appealing component is the development of generic a posteriori error estimators. These could find wide acceptability among scientists who do not have the mathematical background that is necessary to the construction of the traditional a posteriori estimators. A judicious balance will be struck between algorithm and code development, on one hand, and the rigorous analysis of their properties, on the other. Since the dawn of the modern era, scientists have been successful in describing natural phenomena by using partial differential equations (PDE's) as models. The Einstein equations that describe phenomena at the largest (cosmic) scales, the Schrodinger equation that describes phenomena at the smallest (atomic) scales and the Navier-Stokes equations that are used to model a plethora of fluid flows are but a few of such PDE's without which the modern world would not be what it is today. Yet such PDE's are almost impossible to solve exactly and since the beginning, scientists have resorted to numerical calculations to approximate the unknown solutions. Naturally, advances in numerical techinques must keep pace with advances in PDE's and quite recently adaptive numerical techniques have emerged as a very promising tool in tackling even the most difficult approximation tasks. In carrying out the research projects outlined in this proposal, the P.I. will develop novel and promising adaptive numerical algorithms, analyze their properties and implement them on state of the art computers. These methods and computer codes will become part of the arsenal of tools enabling engineers, physicists and mathematicians to understand the phenomena described by the particular PDE's they are using. For example, one research activity proposed herein is to numerically investigate the ways in which a malignant tumor changes shape. Finally, in maintaining the essential tradition of training the next generation of teachers and researchers, a graduate student will participate in these projects in partial fulfillment of his Ph.D. degree.