A discontinuous Galerkin finite element method for directly solving the Hamilton–Jacobi equations

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

“…The second order local solver developed in this paper is based on the second type discontinuous Galerkin method [9] reviewed in the introduction. The main idea will be presented for the cases with rectangular meshes.…”

“…Following the idea of the method developed in [9], a numerical scheme for (1.1) can be formulated as follows: find φ h ∈ V 1 h , such that…”

“…This method is later reinterpreted and simplified in [30] by using the piecewise curl-free solution space in DG method. In another recent work on DG method for H-J equation [9], the solution φ is solved directly. This method is more desirable in the sense that the derived hyperbolic system for ∇φ in [24,30] involves more unknown functions than the original scalar H-J equation which is solved directly in [9].…”

“…In another recent work on DG method for H-J equation [9], the solution φ is solved directly. This method is more desirable in the sense that the derived hyperbolic system for ∇φ in [24,30] involves more unknown functions than the original scalar H-J equation which is solved directly in [9]. Moreover the solutions of H-J equations generally are smoother than those of the conservations laws.…”

“…Based on the DG discretization in [9], in this paper, we develop a second order fast sweeping method for solving Eikonal equations by incorporating the causality property of these equations into the DG local solver. The resulting local nonlinear system in the Gauss-Seidel iterations is a simple quadratic system and can be solved explicitly.…”