An a posteriori error estimate for a semi-Lagrangian scheme for Hamilton--Jacobi equations (Q1410495)

The authors use a semi-Lagrangian scheme to fully discretize the Hamilton-Jacobi-Bellman equation \[ \mu v(x) + \max_{a\in A}\{-b(x,a)\cdot \nabla v(x) -c(x,a)\}, \quad x \in {\mathbb R}^n, \] where \(\mu > 0, A\) is a compact subset of \({\mathbb R}^m\), \(b: {\mathbb R}^n \times A \to {\mathbb R}^n\) and \(c: {\mathbb R}^n \times A \to {\mathbb R}\) are two given functions satisfying certain conditions. They prove a posteriori error estimate for for this scheme and show that this estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution. It is mentioned that the Hamilton-Jacobi equation \(v(x) + H(\nabla v(x)) = 0\), \(x \in {\mathbb R}^n\) with convex Hamiltonian can be written in the form of the above equation