Half-relaxed limits of the monotone diagonal-diffusion HJB scheme

The row weights in [F1] give $\mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}+n{\tau }_{k}F\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}+TF\mathrm{.}$ Multilinear interpolation is a convex combination, so the same bound holds for the space-time extensions. They are locally bounded and their half-relaxed limits have the semicontinuity and order properties in [F6].

Let $\bar{U}-\varphi$ have a local maximum at $(x_0,t_0)$, with $0<t_0<T$ and smooth $\varphi$. Add $\epsilon (\mathrm{\mid }x-x_0{\mathrm{\mid }}^4+\mathrm{\mid }t-t_0{\mathrm{\mid }}^4)$ to $\varphi$; this makes the contact strict without changing its derivatives used by the PDE at $(x_0,t_0)$, as in [F4]. Select nodal local maxima $(x_k,t_k)$ from [F3] and put $c_k=U_{i_k}^{n_k}-\varphi (x_k,t_k)$. All coordinate neighbors used by the explicit stencil at the previous time lie in the fixed contact cylinder for sufficiently large $k$, because their distance to $(x_k,t_k)$ is at most $h_{k,\max }+{\tau }_k$. Thus $U_j^{n_k-1}\le \varphi (x_j,t_k-{\tau }_k)+c_k$ at every stencil node. Monotonicity and shift invariance in [F1] give $U_{i_k}^{n_k}=S_k(U^{n_k-1}{)}_{i_k}\le S_k(\varphi (\cdot ,t_k-{\tau }_k)+c_k{)}_{i_k}=S_k(\varphi (\cdot ,t_k-{\tau }_k){)}_{i_k}+c_k\mathrm{.}$ At the contact node $U_{i_k}^{n_k}=\varphi (x_k,t_k)+c_k$, so the normalized residual is nonpositive. Passing to the limit by [F2] yields ${\varphi }_t-{\max }_a\{{\sum }_j({\alpha }_j^a{\varphi }_{x_j{x}_j}+b_j^a{\varphi }_{x_j})+f^a\}\le 0$ at $(x_0,t_0)$.

At a local minimum of $\underset{‾}{U}-\varphi$, subtract $\epsilon (\mathrm{\mid }x-x_0{\mathrm{\mid }}^4+\mathrm{\mid }t-t_0{\mathrm{\mid }}^4)$ from $\varphi$ to make the contact strict without changing its tested derivatives, as in [F4]. The minimizing nodes in [F3] give $U_j^{n_k-1}\ge \varphi (x_j,t_k-{\tau }_k)+c_k$ on the stencil. Order preservation reverses the update inequality, so the residual is nonnegative and its limit gives the supersolution inequality. By [F4], both inequalities extend from smooth tests to all tests in the declared $C^{1,2}$ class.

The initial-trace estimate [F5] and uniform continuity of $u_0$ show that every space-time sequence with mesh tending to zero and time tending to zero has values tending to $u_0$ at its limiting spatial point; the spatial interpolation adds at most the modulus of $u_0$ at distance $\sqrt{d}{h}_{k,\max }$. Therefore the upper and lower relaxed limits both have the prescribed relaxed initial trace. Boundedness and semicontinuity were proved in step 1.1, completing the stability statement.