L1 convergence of a SL scheme for the eikonal equation with discontinuous coefficient

Abstract : We consider the stationary eikonal equation where the coefficients are allowed to be discontinuous. The discontinuities must belong to a special class for which the notion of viscosity solutions in the sense of Ishii is suitable. We present a semi–Lagrangian scheme for the approximation of the viscosity solution also studying its properties. The main result is an a-priori error estimate in the L 1-norm. In the last section, we illustrate some tests and applications where the scheme is able to compute the right solution. 1. Introduction. In this paper we study the following boundary value problem. Let Ω ⊂ R N be an open bounded domain with a Lipschitz boundary ∂Ω, we consider the Dirichlet problem c(x)|Du(x)| = f (x) x ∈ Ω, u(x) = g(x) x ∈ ∂Ω, (1) where f , c and g are given real functions defined on Ω. We focus our attention on the case where f is positive, Borel measurable and possibly discontinuous. In the most classical case, where c(x) is constantly equal to one, f (x) ≡ 1 and g(x) ≡ 0, we get the eikonal equation giving the characterization of the distance from ∂Ω. In other applications, e.g. in geometrical optics, computer vision, control theory and robotic navigation, c and f can vary but have typically a constant sign (e.g. positive). It is worth to note that in the study of many problems motivated by real world applications a discontinuous f and/or a degenerate c can appear in a natural way. In fact, one can easily imagine that the velocity of a front in a medium is affected by the physical properties of the medium and can be discontinuous if the medium is stratified by different materials. In the famous Shape-from-Shading problem the right-hand side is f (x) = [(1−I 2 (x))/I 2 (x)] 1/2 where I is the brightness of the image. Depending on the shape of the object represented in the image I can be discontinuous. Another motivation to deal with discontinuous Hamiltonians comes directly from
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HYP2012, 2012, Padova, Italy. AIMS, pp.559-567, 2012, Proceedings of the HYP2012 International Conference on Theory, Numerics and Applications of Hyperbolic Problems
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Adriano Festa, Maurizio Falcone. L1 convergence of a SL scheme for the eikonal equation with discontinuous coefficient. HYP2012, 2012, Padova, Italy. AIMS, pp.559-567, 2012, Proceedings of the HYP2012 International Conference on Theory, Numerics and Applications of Hyperbolic Problems. 〈hal-01122925〉

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