NUMERICAL SCHEMES FOR THE HAMILTON-JACOBI AND LEVEL SET EQUATIONS ON TRIANGULATED DOMAINS

Borrowing from techniques developed for conservation law equations, nu merical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The firs t scheme is a provably monotone discretization for the H-J equations. Unfortunately, the basic scheme lacks Lipschitz continuity of the nume rical Hamiltonian. By employing a ''virtual'' edge flipping technique local Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on th e weaker concept of positive coefficient approximations for homogeneou s Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximat ions is considered. These schemes are stabilized via a least-squares b ilinear form. The Petrov-Galerkin schemes do not possess a discrete ma ximum principle but generalize to high order accuracy. Discretization of the level set equation also requires the numerical approximation of a mean curvature term. A simple mass-lumped Galerkin approximation is presented in Section 6 and analyzed using maximum principle analysis. The use of unstructured meshes permits several forms of mesh adaptati on which have been incorporated into numerical examples. These numeric al examples include discretizations of convex and nonconvex forms of t he H-J equation, the Eikonal equation, and the level set equation. (C) 1998 Academic Press.