Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form {Mathematical expression} {Mathematical expression} for all e in the convex support cone of the solution given by {Mathematical expression} {Mathematical expression} where {Mathematical expression} Such inequalities arise in the characterization of saddle-point payoffs u in two person differential games with stopping times as strategies. In this case, H is the Hamiltonian in the formulation. A numerical scheme for approximating u is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence to u of order O(h^1/2) is demonstrated in the L²(0, T; H¹(Ω)) norm, where h is the maximum diameter of a given triangulation.