In this proposal, the investigator studies and develops a new, emerging generation of discontinuous Galerkin methods characterized by being easier to implement, by having enhanced stability and convergence properties, and by displaying an improved flexibility for handling arbitrarily-shaped domains. The investigator will focuses his effort in four particular problems. The first is in the area of fluid flow and consists in establishing a general theory of very competitive numerical methods for the incompressible Navier-Stokes equations. The second is in the area of continuum mechanics and consists in the study of optimally convergent methods for fourth-order problems in order to pave the way to the devising of numerical methods for non-linear shells. The third is in the area of non-linear conservation laws and consists in the introduction of new techniques geared towards overcoming the two main difficulties that have dragged down for more than a decade the development of efficient, high-order accurate methods for these useful equations. The last is in the area of techniques for handling curved boundaries and consists in replacing the traditional paradigm of meshing the domain with high accuracy by the new approach of using a very simple mesh of a box containing the domain and employing special approximation techniques near its border.
The computer simulation of physical phenomena is a highly valued tool of practical interest in a wide variety of applications in Engineering and Physics. The investigator studies an emerging and promising technique of carrying out these simulations with highly accurate and more efficient algorithms for a wide range of problems of practical interest. They include many applications to Aerospace and Mechanics (incompressible fluid flow, subsonic and supersonic flow) as well as to Civil Engineering (solid structures).