This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5),
The main part of this project will focus on developing and analyzing discontinuous Galerkin (DG) methods for problems arising in structural mechanics and fluid flow. In particular, the P.I. will analyze hybridizable discontinuous Galerkin (HDG) methods for plate bending problems, elasticity equations and convection-diffusion equations. One advantage of HDG methods is that they can approximate all the variables of interest in an optimal way while using equal-order approximations for all the variables. More importantly, many of the global degrees of freedom can be eliminated by the use of Lagrange multipliers, making the final linear system smaller than linear systems arising in standard DG methods. Another component of the project is the investigation of DG methods for multiscale problems. The P.I. hopes to develop higher-order DG methods using non-polynomial basis functions. A final project will be answering theoretical questions in finite elements. The P.I. will prove pointwise error estimates for finite element methods applied to the Stokes problem on general convex polyhedral domains. Then, the P.I. will prove error estimates for higher-order streamline diffusion methods on layer-adapted meshes.
Numerical simulations play a central role in modern engineering. For example, they are crucial in the design of airplanes, automobiles, and oil platforms, to name a few. They allow industries to test structures using computers without ever building an actual physical model. One of the reasons this is possible is that very efficient and reliable numerical methods have been developed over the years. However, to meet new computational challenges, researchers are working on improving existing algorithms and on the development of new competitive ones. In this project, the P.I. will work on developing a new, promising family of numerical methods called hybridizable discontinuous Galerkin methods. In order to gain a deeper understanding of these numerical methods and related ones, the P.I. will also investigate mathematical aspects of such methods.
