Let Ω be a bounded polygonal domain in . We consider a model problem defined as follows.
For given , find a scalar function ϕ on Ω being a minimizer of the cost functional
(1)
over the convex set K defined by
where the Sobolev space is defined as follows:
Variational inequalities might be separated into two main groups as elliptic and parabolic variational inequalities. Glowinski studies these sorts of inequalities in [3] in detail. He considers the elliptic variational inequalities (EVI) as the first kind and second kind EVI and defines those in a functional context as follows.
Let V denote a real Hilbert space with the inner product and the associated norm . is a dual space of V, is a bilinear, continuous and V-elliptic form on , continuous linear functional, K is a closed convex nonempty subset of V, is a convex lower semi-continuous functional. The first and second kind EVI are typically defined in the following way.
The first kind EVI: find such that ϕ is a solution of the problem
The second kind EVI: find such that ϕ is a solution of the problem
The next lemma sets up the connection between optimization and VI problems.
Lemma 1 [3]
Let be a symmetric continuous bilinear V-elliptic form. Let and be a convex lower semi-continuous proper functional. Let
Then the minimization problem, find ϕ such that
has a unique solution which is characterized by
Proof The proof can be seen on p.7 of [3]. □
It is clear that we can define the following in terms of the notations of Lemma 1:
We will approximate (1) with a finite element method introduced in [3]. Assume that Ω is a polygonal domain of . Consider a triangulation of Ω in the following sense: is a finite set of triangles T such that
Here denotes the inner part of the corresponding triangle. Furthermore, for all and , exactly one of the following conditions must hold:
(i) ,
(ii) and have only one common vertex,
(iii) and have only a whole common edge.
h is the length of the largest edge of the triangles in the triangulation. Define as a space of polynomials in and of degree less than or equal to k, and
The space is approximated by the family of subspaces with or , where
It is obvious that the are finite dimensional. Then the space K is approximated by
Notice that for are closed convex nonempty subsets of .
With these settings, the solution is approximated by
(2)
or equivalently,
Using the augmented Lagrangian multipliers method, we find a discrete solution of (1) as follows. First, let us introduce the Lagrange functional
For , an augmented Lagrangian is defined by
(3)
Augmented Lagrangian multipliers methods for VI problems have been introduced by Glowinski and Marrocco (see [8]). Theorem 2.1 on p.168 in [3] guarantees the existence of a solution of this optimization problem.
Let us do notice that the first component of (2) is then the solution of the original problem (1). Using the techniques of the variational calculus, we can write that
or equivalently,
We can describe the solution algorithm as follows:
(i) Choose an initial iterate and ;
(ii) Solve the linear problem , ;
(iii) Update on each cell;
(iv) Set and go back to (ii).