Adaptive Discontinuous Galerkin Methods and Applications

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Scientific computing and especially computational mathematics are recognized as crucial to the advancement of science. By replacing expensive and sometimes even impossible laboratory experiments (e.g., supernova explosions), numerical simulations have grown steadily in importance as effective alternatives to these. Therefore, the development of state of the art algorithms and codes, with the concomitant increase in confidence in their predictions, is important to progress in all these fields. The research program in this project aims at the development, analysis, and computer implementation of numerical methods designed to approximate the solutions of some partial differential equations that govern key phenomena in the fields of applied mathematics, biomedicine, engineering, and physics. In particular, this research program will be applied to generate efficient numerical algorithms for the effective simulation of the functioning of the heart by focusing on its electro-mechanical nature and the resulting fluid flow patterns. Another application of importance to society will be to apply the newly developed algorithms to the simulation of the Earth's climate by studying the influence of various chemicals including greenhouse gases, as well as pollutants such as aerosols, dust and black carbon. The knowledge gained will be communicated to other researchers through seminars and conferences. New courses will be developed and students will participate in all phases of the project. The latter aspect is crucial in developing the new generation of researchers.The numerical methodologies described in this proposal namely the finite element method and in particular the discontinuous Galerkin method have been studied for several decades now. Yet there is ample need for growth in both the analytical and application arenas. Research on traditional as well as recently introduced a posteriori error estimates will be conducted extending these estimates to classes of equations that had stayed beyond the reach of such methods. These will lead to improved and flexible adaptive methods that will be applied to solving problems of practical importance. An additional application of adaptive methods will be to answer questions of analytical nature concerning the existence, uniqueness, and stability properties of solitary wave solutions of nonlinear dispersive equations. The crucial problem of solving the resulting systems of linear and nonlinear equations will form an integral component of the proposed research with multilevel domain decomposition and multigrid methods forming the backbone of the thrust in this context. Implementation on massively parallel computers will be pursued actively given that most of the problems studied involve three spatial dimensions as well as time.