In recent years, the Hamilton-Jacobi and levelset equations have been successfully applied in the modeling of a large number of problems arising in image processing, computer vision, fluid mechanics, obstacle navigation
(http://www.nas.nasa.gov/~barth/images/obstacle.gif), path planning
(http://www.nas.nasa.gov/~barth/terrain/mars.gif), and moving interfaces
(http://www.nas.nasa.gov/~barth/movies/hj_movie.mpg). For numerous further examples, see the books by Sethian (1996) and Shapiro (2001) as well as the review article of Osher and Fedkiw (2000).
In the present talk, we consider the fast high order accurate numerical approximation of the eikonal
equation on triangulated manifolds using linear, quadratic, and cubic isoparametric (curved) element approximation (http://www.nas.nasa.gov/~barth/images/eikonal2.gif). Previous methods based on explicit space marching have been limited to low order accuracy and barrier theorems exist for the linear hyperbolic counterpart of the eikonal equation which confirm this restriction. We then consider a new *scalar* form of the stabilized discontinuous Galerkin discretization for the eikonal equation and show how fast solution methods are produced which circumvent the accuracy barrier theorems produced by explicit marching methods. The discontinuous Galerkin method generalizes to high order accuracy on irregular meshes and readily permits the generalization to piecewise isoparametric element approximation of curved (possibly non-orientable) manifolds. Numerical eikonal examples containing obstacles, variable refractive index materials, and curved surfaces are shown throughout the talk to demonstrate the generality of the techniques.
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