Title: An adaptive method for Hamilton-Jacobi equations


			  Bernardo Cockburn
		      Department of Mathematics
		       University of Minnesota
                 127 Vincent Hall, 206 Church St. S.E.
                     Minneapolis, MN 55455 USA
                    Email: cockburn@math.umn.edu
                Web page: www.math.umn.edu/~cockburn


Abstract:
  In this talk, we present the first adaptive method for finding
approximations for Hamilton-Jacobi equations whose $L^\infty$-distance to
the viscosity solution is no bigger than a prescribed tolerance. The
method proceeds as follows. On any given mesh, the approximate solution is
computed by using a well known monotone scheme; then, the quality of the
approximation is tested by using an approximate a posteriori error
estimate. If the error is bigger than the prescribed tolerance, a new mesh
is computed by solving a suitably devised differential equation. A
thorough numerical study of the method is performed which shows that
rigorous error control is achieved, even though only an approximate a
posteriori error estimate is used; the method is thus reliable.
Furthermore, the numerical study also shows that the method is efficient
and that it has an optimal computational complexity. These properties are
independent of the value of the tolerance.