An Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Hamilton–Jacobi Equations

In this paper, an arbitrary Lagrangian–Eulerian local discontinuous Galerkin (ALE-LDG) method for Hamilton–Jacobi equations will be developed, analyzed and numerically tested. This method is based on the time-dependent approximation space defined on the moving mesh. A priori error estimates will be stated with respect to the L∞0,T;L2Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^{\infty }\left( 0,T;\mathrm {L}^{2}\left( \Omega \right) \right) $$\end{document}-norm. In particular, the optimal (k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document}) convergence in one dimension and the suboptimal (k+12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+\frac{1}{2}$$\end{document}) convergence in two dimensions will be proven for the semi-discrete method, when a local Lax–Friedrichs flux and piecewise polynomials of degree k on the reference cell are used. Furthermore, the validity of the geometric conservation law will be proven for the fully-discrete method. Also, the link between the piecewise constant ALE-LDG method and the monotone scheme, which converges to the unique viscosity solution, will be shown. The capability of the method will be demonstrated by a variety of one and two dimensional numerical examples with convex and noneconvex Hamiltonian.