= Solution
Take $F=(F^*)^*$ with the usual proper closed convex assumptions, so the <Fenchel-Moreau theorem> identifies its conjugate with the supplied $F^*$. Put $z^k=u^k-\tau D^Tv^k$. The first update is $w^{k+1}=\operatorname{prox}_{F^*/\tau}(z^k/\tau)$. By <Moreau decomposition>, the second update is
$$
u^{k+1}=z^k-\tau\operatorname{prox}_{F^*/\tau}(z^k/\tau)=\operatorname{prox}_{\tau F}(z^k).
$$
Also $\tau(D^Tv^k+w^{k+1})=u^k-u^{k+1}$, so the argument in the final update contains $u^{k+1}-\tau(D^Tv^k+w^{k+1})=2u^{k+1}-u^k$. Eliminating the auxiliary variable gives
$$
\boxed{\begin{aligned}
u^{k+1}&=\operatorname{prox}_{\tau F}(u^k-\tau D^Tv^k),\\
v^{k+1}&=\operatorname{prox}_{\sigma G}\left(v^k+\sigma D(2u^{k+1}-u^k)\right).
\end{aligned}}
$$
This is the <primal-dual hybrid gradient method> with extrapolation parameter one and the primal update performed first. Its <saddle point> function is $F(u)+\langle Du,v\rangle-G(v)$, corresponding to the primal objective $F(u)+G^*(Du)$.
For proper <lower semicontinuous> <convex functions> and a nonempty saddle-point set in these finite-dimensional spaces, the standard <convergence of primal-dual hybrid gradient> theorem gives the sufficient parameter condition
$$
\boxed{\tau>0,\qquad\sigma>0,\qquad\tau\sigma\|D\|_2^2<1.}
$$
Here $\|D\|_2$ is the <operator norm>, or largest <singular value>. For $D\ne0$, one possible choice is $\tau=\sigma=\eta/\|D\|_2$ with $0<\eta<1$; for $D=0$ any positive steps satisfy the condition. The existence assumption is necessary: step sizes alone cannot guarantee convergence to a saddle point that does not exist.
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