Discontinuous galerkin methods for flow and electro-magnetic problems

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Discontinuous Galerkin methods are specialized numerical methods for the computer simulation of complex phenomena in the sciences and engineering. Although the first discontinuous Galerkin method was devised back in 1973, these methods experienced a significant development during the nineties which brought them to the mainstream of computational fluid dynamics. In recent years, these methods are being applied to many other problems of engineering interest including electro-magnetics and structural mechanics. The recent success of discontinuous Galerkin methods can be attributed to their flexibility and versatility. These methods can deal robustly with partial differential equations of almost any kind, as well as with equations whose type changes within the computational domain. They are ideally suited for multi-physics applications and for problems with highly varying material properties in complex geometries. Moreover, discontinuous Galerkin methods can easily handle irregularly refined meshes and variable approximation degrees; a property referred to as hp-adaptivity.The long-term objectives of this proposal are the development, analysis and implementation of discontinuous Galerkin finite element methods for the improved simulation of problems in computational fluid mechanics and electro-magnetics. In particular, we plan to address some of the most relevant issues of the current development of discontinuous Galerkin methods: the design of new methods that have fewer globally coupled degrees of freedom, the development of exactly divergence-free methods, the analysis of new and robustly convergent adaptive algorithms, and the devising of new solvers and implementation techniques. We further plan to develop a flexible simulation code for problems in incompressible fluid flow and electro-magnetics, which incorporates modern developments in computational mathematics, such as hp-adaptivity, efficient iterative solvers, and high-order time-stepping.