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



Numerical analysis of parabolic obstacle problem
There's an overview of available schemes in chapter III of


*Roland Glowinski*, MR 737005 [**Numerical methods for nonlinear variational problems**](http://dx.doi.org/10.1007/978-3-662-12613-4), ISBN: 0-387-12434-9.


(of which there is also reprint from 2008). The schemes are presented and a few references for their behaviour are given. In particular, this book references chapter 6 of


*Roland Glowinski, Jacques-Louis Lions, and Raymond Trémolières*, MR 1333916 [**Numerical analysis of variational inequalities**](http://www.ams.org/mathscinet-getitem?mr=1333916), ISBN: 0-444-86199-8.


which might take you further, even though at a first glance I didn't see the explicit Euler method covered.


