Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part III: Duality and Penalty Finite Element Methods (Journal of Computational Mathematics, 2021)
The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints can be treated by the duality and penalty methods and then an unconstrained problem can be obtained. In this paper we develop methods applying the finite element methods to compute solutions of linear-quadratic $N$-person games using duality and penalty formulations. The calculations are efficient and accurate. When a (4,1)-system of Hermite cubic splines are used, our numerical results agree well with the theoretical predicted rate of convergence for the Lagrangian. Graphs and numerical data are included for illustration.