Level Set Methods for Adaptive Grids.
It has been well documented that mesh adaptation is of critical importance in the numerical solution of partial differential equations. The accuracy and efficiency improvements often warrant the extra work entailed in implementing an adaptive mesh.
A promising direction in the field of adaptivity are the so-called moving mesh methods. In such methods, a mesh equation is solved (often simultaneously with the original differential equation) for node velocities, which move/keep the nodes concentrated in regions of rapid variation of the solution.
In practice, however, most moving mesh methods suffer from an effect known as "mesh crossing". These typically arise because of some difference approximations when calculating the mesh velocities.
My poster discusses a new moving mesh method based on Level Set Ideas. Such methods eliminate mesh crossings entirely. They also allow for topology change in the solution or domain. I will then show some numerical solutions of the Porous Medium Equation (PME) and Fisher's Equation using this new moving mesh method.