I want to solve a parabolic obstacle problem, written as a variational inequality: For almost all $t\in [0,T]$

\begin{align*} \langle u'(t), v - u(t)\rangle +a(u(t),v-u(t)) \geq \langle f(t),v-u(t)\rangle \quad \forall v \in K \end{align*}

with $K = \{v \in H^1_0(\Omega) ~\vert ~ v \geq \chi ~ \text{ f.a.a }~ x \in \Omega\}$ and $u(0) = u_0$. Now we will discretize this inequality in time using, for instance, the explicit Euler Scheme. After this we need to solve an elliptic problem in each timestep. This will be done by a primal-dual-active-set method following Bartels book "Numerical methods for Nonlinear Partial Differential Equations". Can someone give me a hint or literature, how to prove the convergence of this "method" to a solution of the obstacle problem?

Thanks in advance, FFoDWindow