This project is devoted to devising and studying, theoretically as well as computationally, efficient methods for numerically solving problems modeled by partial differential equations. We propose to investigate how to overcome several difficulties raised by the application of the so-called discontinuous Galerkin method to a variety of problems of practical interest. The applications are to the evolution of surfaces and to navigation of robots in arbitrary terrains (Hamilton-Jacobi equations), to incompressible fluid flow (Navier-Stokes equations), to structural mechanics with special emphasis on shells (elasticity equations), and to Darcy flow in porous media (second-order elliptic equations).
For each of the above problems, we consider the problem of how to obtain highly accurate computer simulations. In many of these problems (fluid flow and structural mechanics), the emphasis will be put on the development of a recently discovered class of discontinuous Galerkin methods more robust and efficient than previously known methods. In others (evolution of surfaces and navigation of robots), the effort will be concentrated in the efficient control of the quality of the simulation. Indeed, since the exact solution of these complex problems is not known, to guarantee a given accuracy of the simulation, special techniques have to be devised in order to assess its quality. Moreover, these techniques can be employed to automatically let the computer know when and where to increase or decrease the computational effort to obtain the simulation; in this way, the efficiency of its computation is significantly enhanced. Discontinuous Galerkin methods are particularly well-suited for this approach.
