Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton–Jacobi equations

Abstract

In this note, we reinterpret a discontinuous Galerkin method originally developed by Hu and Shu [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690] (see also [O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434]) for solving Hamilton–Jacobi equations. With this reinterpretation, numerical solutions will automatically satisfy the curl-free property of the exact solutions inside each element. This new reinterpretation allows a method of lines formulation, which renders a more natural framework for stability analysis. Moreover, this reinterpretation renders a significantly simplified implementation with reduced cost, as only a smaller subspace of the original solution space in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is used and the least squares procedure used in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is completely avoided.