L-minimization Methods for Hamilton-jacobi Equations: the One-dimensional Case

The goal of the present paper is to investigate the approximation properties of a new class of L-minimization techniques for approximating stationary HamiltonJacobi equations in one space dimension. Most approximation algorithms of Hamilton-Jacobi equations are based on monotonicity and Lax-Friedrichs approximate Hamiltonians, see e.g. Kao, Osher, and Tsai [13]. Monotonicity is very often invoked to prove convergence of low-order approximations, see e.g. Crandall and Lions [6], Barles and Souganidis [2]. In this spirit, Abgrall [1] proved convergence for a class of first-order schemes on meshes composed of triangles using a monotonicity-based argument by Crandal and Lions [6]. For higher-order approximations, limiters must be brought into the game as monotonicity cannot be preserved. For instance, second-order MUSCL-type finite difference approximations have been shown to converge to viscosity solutions by Lions and Souganidis [17], see also Osher and Shu [19] for higher-order discretizations. We refer to Osher and Fedkiw [18] and Sethian [21] for reviews of the approximation literature of Hamilton-Jacobi equations. In the present paper we take a radically different point of view by formulating the discrete problem as a minimization in L(a, b). The motivation behind this approach is based on observations made in [8] that L-minimization is capable of selecting viscosity solutions of transport equations equipped with ill-posed boundary conditions. This fact has indeed been proved in [10] in one space dimension. This encouraged us to build a research program in this direction and the purpose of the present work is to show that indeed L-minimization is a viable technique. The idea of using L to construct nonlinear approximations of