# Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations

L. Grüne, F. Camilli:
**Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations**

*SIAM Journal on Numerical Analysis*
**38**,
1540 - 1560,
2000

DOI:
10.1137/S003614299834798X

Keywords: singular Hamilton-Jacobi equations; maximal solution; regularization; numerical approximation; degenerate Hamilton-Jacobi equations; error bound; viscosity solution; eikonal equation; shape-from-shading problem

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## Abstract:

In this paper we study an approximation scheme for a class of Hamilton-Jacobi problems for which uniqueness of the viscosity solution does not hold. This class includes the Eikonal equation arising in the Shape from Shading problem. We show that, if an appropriate stability condition is satisfied, the scheme converges to the maximal viscosity solution of the problem. Furthermore, we give an estimate for the discretization error.