Overlapping Nonmatching Grid Method for the Ergodic Control Quasi Variational Inequalities ()
1. Introduction
The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping sub domains.
The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the sub domain.
In this paper, we are interested in the error analysis in the maximum norm for the obstacle problem in the context of overlapping nonmatching grids: we consider a domain Ω which is the union of two overlapping subdoains where each sub domain has its own triangulation. This kind of discretizations is very interesting as they can be applied to solving many practical problems which cannot be handled by global discretizations. They are earning particular attention of computational experts and engineers as they allow the choice of different mesh sizes and different orders of approximate polynomials in different sub domains according to the different properties of the solution and different requirements of the practical problems.
We study a new approach for the finite element approximation for the ergodic problem where the obstacle is related to a solution. We consider a domain which the union of two overlapping sub-domains where each sub domain has its own generated triangulation. The grid points on the sub-domain boundaries need not much the grid points from the other sub-domains. Under a discrete maximum principle, we show that the discretization on each sub-domain converges quasi-optimally in the 
norm.
In the first section we study the Schwarz method for the ergodic control Quasi-variational inequalities; we state the continuous alternating Schwarz sequence for quasivariational inequalities, and define their respective finite element counterparts in the context of overlapping grids.
In Section 2, we give a simple proof for the main result concerning error estimates in the 
norm for the problem studied, taking into account the combination of Jinping Zeng & Shuzi Zhou [2] geometrical convergence and P. L. Lions, B. Perthame [3] quasi-variational inequalities and ergodic impulse control.
2. Schwarz Method for the Ergodic Control Quasi-Variational Inequalities
We begin by down a classical results related to ergodic control quasi-variational inequalities [1-18].
It is well known that impulse control problems for reflected diffusion process may be solved by considering the solution of quasi variational inequalities (QVI) (see Bensoussan [4], A. Bensoussan and J. L. Lions [5]). A typical example is the following:
(1)
where Ω is given bounded smooth open set in
, 
, f is given function, the cost function 
represents the obstacle of impulse control defined by:
(2)
where k is a positive number, M is an operator defined on 
and assumed to map into itself, that is,
whenever 
(3)
; and (.,.) denotes the inner product on Ω.
It has been proved that the long run average cost for this problem solves the ergodic QVI. More precisely, denoting by:
and
.
P. L. Lions and B. Perthame [3] proved that the solution 
of the QVI :
(4)
converges to the solution of the ergodic control QVI :
(5)
As stated in the following theorem.
Theorem 1 [1]: As 
goes to
,
converges uniformly in 
to some constant
, and 
converges uniformly in 
and strongly in 
to
. Moreover 
is the unique solution of the quasi variational inequality of the ergodic control problem (5).
Problem Position
Let 
be fixed in the open interval 
and set. Then, one can easily see that problem (1) is equivalent to the following QVI:
(6)
where
, and 
(7)
Thanks to [5], (1) or (6) has a unique solution. Also, notice that, as the bilinear form (7) is independent of
, the left hand side of (6) is independent of 
too.
3. The Schwarz Method for the Obstacle
We decompose 
into two overlapping polygonal sub domains 
and
, such that
(8)
and 
satisfies the local regularity condition
(9)
We denote by 
the boundary condition of 
and 
the intersection of 
and 
is assumed to be empty.
3.1. The Schwarz Sequences for Problem (4)
We denote by:
and 
(10)
Choosing 
such that 
the unique solution of:
(11)
We respectively define 
the alternating Schwarz sequences on 
such that:
(12)
where
.
3.2. The Continuous Schwarz Sequences for Principal Problem
We consider the following problem:
(13)
Choosing
, solution of:
(14)
We respectively define the alternating Schwarz sequences 
on 
such that: 
and;
(15)
(16)
(17)
Lemma 1 [3]: for each 
4. The Discret Problem
We suppose for simplicity that 
is polyhedral. Let 
be a regular, quasi uniform triangulation of 
into nsimplexes of diameter less than h.
We denote by 
the standard piecewise linear finite element espace, we consider the discrete variational inequality:
(18)
Thanks to ([6]), QVI (18) has a unique solution.
4.1. The Discrete Maximum Principle [7]
We assume that the matrix A with generic coefficient is a M-matrix.
(19)
As for the continuous problem, it is easy to see that
, the solution of (18), is also solution to the following (QVI):
(20)
Theorem 2 [1]: Let 
and 
(the discrete solution). Then, there exists a constant independent of 
the both and h such that
Theorem 3 [1]: Under conditions of theorem 1 and 2, we have
Note that 
and 
is the finite element approximation of 
and 
respectively.
4.2. The Discrete Schwarz Sequences
Let 
be the space of continuous piecewise linear function on 
which vanish on
.
For
, we define
where 
denotes the interpolation operator on
.
For
. Let 
be a standard regular finite element triangulation in
, 
being the mesh size.
We suppose that the two triangulations are mutually independent on 
a triangle belonging to one triangulation does not necessarily belong to the other.
Choosing
, such that 
is a solution of the following inequation:
(21)
We define the alternating Schwarz sequences 
on 
such that:
(22)
And 
on 
such that:
(23)
Notation: We will adapt the following notations:
(24)
5. 
-Error Analysis
Lemma 2 [8]: 
is M-matrix such that
then exists constants 
such that
and 
(25)
Theorem 4 [8]: 
produced by Schwarz alternating method converges geometrically to 
the solution of obstacle problem, more precisely, there exist 
which depend only respectively of 
and
, such that
(26)
Theorem 5: Let 
Then, there exist two constants C and
, independent of both h and n such that:
and
.
Proof:
We obtain
.
The case i = 2 is similar.
Theorem 6: Let
and
, the discretes alternating Schwarz sequences, we have:
Proof:
1)
For: 
and
.
The case i = 2 is similar.
.
2)
Hence
: for 
, For: 
and
.
The case i = 2 is similar.
.