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

Abstract

In this paper we consider two-sided parabolic inequalities of the form

$$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$

((li))

$$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$

((lii))

for alle in the convex support cone of the solution given by

$$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$

$$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$

((liii))

where

$$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$

Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.