CURRENT RESEARCH
The Obstacle Problem
The obstacle problem is a variational inequality that finds the equilibrium position of an elastic membrane that passes through the boundary of an open set, that is subjected to a force of density, and lies over an obstacle. A basic summary of the obstacle problem can be found here paper. The image to the left is the solution of the obstacle problem (in this case, the obstacle function is f(x,y)=0).
The DWDG Methods
The dual-wind discontinuous Galerkin (DWDG) methods are a class of discontinuous Galerkin (DG) methods that are formed from DG derivatives that can be found in this paper. The DWDG methods are a class of DG methods that can be penalty-free. This is a big benefit for industrial applications. In my dissertation, I have been able to prove that this method has optimal convergence rates with low regularity of the true solution. The movie to the left is the penalty-less DWDG approximation to the solution given in the image above.