Finite element approximation of the Isaacs equation
This work provides a rigorous numerical analysis for a challenging PDE arising in differential games, but the method is incremental (extending existing techniques).
The authors propose a two-scale finite element method for the Isaacs equation and prove convergence to the viscosity solution as mesh size and approximation parameter tend to zero, with rates under smoothness assumptions.
We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size $h$ whereas the coarse scale $\varepsilon$ is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as $\varepsilon, h\to0$, and $\varepsilon \gtrsim h^{1/2}|\log h|$. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.