A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case

Abstract

A new upper bound is provided for the L ∞-norm of the difference between the viscosity solution of a model steady state Hamilton-Jacobi equation, u, and any given approximation, v. This upper bound is independent of the method used to compute the approximation v; it depends solely on the values that the residual takes on a subset of the domain which can be easily computed in terms of v. Numerical experiments investigating the sharpness of the a posteriori error estimate are given.

A posteriori error estimates for the effective Hamiltonian of dislocation dynamics

“…To this end, we consider approximate solutions of the discrete cell problem (1.8), and the goal of this paper is to provide some a posteriori error estimates between any discrete effective Hamiltonian and the continuous effective Hamiltonian λ. Concerning the effective Hamiltonian for local Hamilton-Jacobi equations, we refer the reader to [1], [8], [17], [22], [25] for error estimates and numerical computations; see also [2], [3] for a posteriori error estimates.…”

Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions

“…For HJ equations of the form (2) we need to calculate rUðxÞ, so we assume the RBF /ðxÞ is well behaved and differentiate (3).…”

“…These do not require augmentation by polynomials when solving (3). In order to achieve better reconstructions we will attempt to optimize the parameter (e.g.…”

“…Having a consistent scheme, in order to obtain a convergence estimate we could construct a doublingvariable function and proceed along the lines of the proof in [3,9,10]. However, we focus on the numerical implementation here.…”

Paraxial Eikonal Solvers for Anisotropic Quasi-P Travel Times

“…We expect similar applications for the algorithm presented here. There are other newly developed high-resolution schemes for Hamilton-Jacobi equations, such as central-difference schemes [24,26], discontinuous Galerkin schemes [1,15], and finite-volume schemes [22]; we plan to test these schemes on the eikonal equations in the seismic exploration setting in the near future. For examples of capturing multivalued travel times and caustics by solving Hamilton-Jacobi equations, see [2,3].…”