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

Abstract: 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 $\mathrm{L}^{\infty}\left(0,T;\mathrm{L}^{2}\left(\Omega\right)\right)$-norm. In particular, the optimal ($k+1$) convergence in one dimension and the suboptimal ($k+\frac{1}{2}$) convergence in two dimensions will be proven for the semi-discrete method, when a local Lax-Friedrichs flux and piecewise polynomial 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.