\documentclass{abstract}
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\title{Discontinuous Galerkin method for Hamilton-Jacobi
        equations and front propagation with obstacles}
\author{Chi-Wang Shu}
\institute{Division of Applied Mathematics
                      Brown University}
\email{shu@dam.brown.edu}

\mymaketitle

\section{Abstract}
In this talk we will first describe a discontinuous Galerkin
(DG) method for solving Hamilton-Jacobi equations, including 
those for front propagation problems.  This method solves the
Hamilton-Jacobi equations directly, without first converting
them to conservation law systems, can be proved to converge
optimally in $L^2$ for smooth solutions, and perform nicely for
viscosity solutions with singularities.  

We then extend the DG 
method to front propagation problems in the presence of 
obstacles.  We follow the formulation of Bokanowski et al. 
leading to a level set formulation driven by $\min(u_t + 
H(x,\nabla u), u-g(x))=0$, where $g(x)$ is an obstacle function.  
The DG scheme is motivated by the variational formulation when 
the Hamiltonian $H$ is a linear function of $\nabla u$, 
corresponding to linear convection problems in presence of 
obstacles. The scheme is then generalized to nonlinear
equations, resulting in an explicit form which is very efficient
in implementation.  Stability analysis are performed for the 
linear case with Euler forward, a second and third order
SSP Runge-Kutta time discretization, and convergence is 
proved for the linear case with Lipschitz continuous and
piecewise smooth data.  Numerical examples are provided to
demonstrate the robustness of the method. Finally, a narrow 
band approach is considered in order to reduce the 
computational cost.  This is a joint work with Yingda
Cheng (the design of the scheme), Tao Xiong (error estimates
for smooth solutions), and Olivier Bokanowski and
Yingda Cheng (front propagation without and with obstacles).
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