# Numerical analysis of parabolic obstacle problem

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

## 1 Answer

There's an overview of available schemes in chapter III of

Roland Glowinski, MR 737005 Numerical methods for nonlinear variational problems, 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, 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.

• Thank you for your answer. Unfortunately he just 'lists' the algorithms, but doesn't provides proves for convergence... – FredTheBread Feb 8 '17 at 18:49
• It gives an overview but also provides references for proofs. I've added the one that seems most important to me to my answer. Now, I have to admit that I don't see the explicit Euler scheme covered there right away, although other schemes are covered. It sounded to me like you were interested in having a starting point were not very much constrained on the precise method that is used. Does this help you? – anonymous Feb 8 '17 at 18:58
• Yeah, I took a brief look into the book and it looks helpful. Thanks! – FredTheBread Feb 8 '17 at 19:04