Computing and Visualization in Science

, Volume 9, Issue 2, pp 57–69

# Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle

• Folkmar Bornemann
• Christian Rasch
REGULAR ARTICLE

## Abstract

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

## Keywords

Hamilton–Jacobi equation Linear finite elements Local variational principle Viscosity solutions Compatibility condition Hopf–Lax formula Eikonal equation Adaptive Gauss–Seidel iteration

## References

1. Alt H.W. (1999). Lineare Funktionalanalysis, third edition. Springer, Berlin Heidelberg New York
2. Bardi M., Capuzzo-Dolcetta I. (1997). Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston MR 99e: 49001
3. Barth T.J., Sethian J.A. (1998). Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. J. Comput. Phys. 145(1):1–40 MR 99d: 65277
4. Crandall M.G., Evans L.C., Lions P-L. (1984). Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282(2):487–502 MR 86a: 35031
5. Evans L.C. (1998). Partial differential equations. American Mathematical Society, Providence MR 99e: 35001
6. Fomel, S.: A variational formulation of the fast marching eikonal solver, Tech. Report 95, pp 127–149, Stanford Exploration Project, Stanford University (1997) sepwww.stanf ord.edu/public/docs/Google Scholar
7. Ishii H. (1987). A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type. Proc. Amer. Math. Soc. 100(2):247–251 MR 88d: 35040
8. Kimmel R., Sethian J.A. (1998). Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. USA 95(15):8431–8435 MR 99d: 65359
9. Lions P.-L. (1982). Generalized solutions of Hamilton-Jacobi equations. Pitman, Boston MR 84a: 49038
10. Li X-G., Yan W., Chan C.K. (2003). Numerical schemes for Hamilton–Jacobi equations on unstructured meshes. Numer. Math. 94(2):315–331 MR 2004b: 65153
11. Osher S., Fedkiw R. (2003). Level set methods and dynamic implicit surfaces. Springer, Berlin Heidelberg New York MR 2003j: 65002
12. Osher S., Sethian J.A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1):12–49 MR 89: 80012
13. Plaum, C., Rüde, U.: Gauß’ adaptive relaxation for the multi- level solution of partial differential equations on sparse grids. Tech. Report SFB-Bericht 342/13/93, Technische Universität München, 1993, www10.informatik.uni-erlangen.de/~ruede/Google Scholar
14. Rockafellar R.T. (1970). Convex analysis. Princeton University Press, Princeton MR 43: 445
15. Rouy E., Tourin A. (1992). A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29(3):867–884 MR 93d: 65019
16. Sethian, J.A.: Theory, algorithms, and applications of level set methods for propagating interfaces. Acta Numer. 5, 309–395 Cambridge University Press, Cambridge (1996) MR 99d: 65397Google Scholar
17. Sethian, J.A.: Level Set Methods and Fast Marching Methods, second ed., Cambridge Monographs on Applied and Computational Mathematics, vol. 3, Cambridge University Press, Cambridge, Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. MR 2000c: 65015 (1999)Google Scholar
18. Sethian J.A., Vladimirsky A. (2000). Fast methods for the eikonal and related Hamilton-Jacobi equations on unstructured meshes. Proc. Natl. Acad. Sci. USA 97(11):5699–5703 MR 2001b: 65100
19. Sethian J.A., Vladimirsky A. (2001). Ordered upwind methods for static Hamilton–Jacobi equations. Proc. Natl. Acad. Sci. USA 98(20):11069–11074 MR 2002g: 65133
20. Sethian J.A., Vladimirsky A. (2003). Ordered upwind methods for static Hamilton–Jacobi equations: theory and algorithms. SIAM J. Numer. Anal. 41(1):325–363 MR 1 974505
21. Tsitsiklis J.N. (1995). Efficient algorithms for globally optimal trajectories. IEEE Trans. Automat. Control 40(9):1528–1538 MR 96d: 49039