In its broader outlines, this research program aims at the development, analysis and computer implementation of numerical methods designed to approximate the solutions of some partial differential equations that have important applications in the fields of engineering and physics. Indeed, elliptic equations, the Navier-Stokes equations and nonlinear wave equations despite having been cultivated for decades, still offer fertile ground for further exploration, for there are still a plethora of unanswered questions and a pressing need for more efficient and faster algorithms. The discontinuous Galerkin method will constitute the core methodology of this effort. While going back to 1973, major interest did not focus on it until the nineties. Today it constitutes one of the most active areas within finite elements if not the whole range of methods for the numerical treatment of partial differential equations. It has not been as extensively explored as the standard Galerkin version, yet what is known so far offers a tantalizing glimpse of its potential. The project will involve various areas at the cutting edge of numerical analysis and scientific computing, in particular, the development of convergent and efficient adaptive methods designed to reduce the run time of the algorithms by finding optimal or quasi-optimal meshes. These adaptive methods will require continuing the work on the development of sharp a-posteriori error estimators designed to identify regions where the solution is varying rapidly. Major efforts will be directed towards pursuing a recently identified strategy for reducing the number of iterations that current adaptive algorithms require in achieving a prescribed level of accuracy.
Scientific computing is recognized as crucial to the advancement of science. As such, the development of state of the art algorithms and codes is important for the progress of technology. Specifically, improved adaptive codes resulting from this project will have impact on a wide range of problems and applications involving fluid flow phenomena, the elucidation of extremely fast chemical reactions by femtosecond lasers, and numerical simulations of supernova explosions.
