L ∞-Error Estimate of Schwarz Algorithm for Noncoercive Variational Inequalities

The Schwarz method for a class of elliptic variational inequalities with noncoercive operator was studied in this work. The author proved the error estimate in L-norm for two domains with overlapping nonmatching grids using the geometrical convergence of solutions and the uniform convergence of subsolutions.


Introduction
More than one hundred years ago, Schwarz algorithms were proposed for proving the solvability of PDEs on a complicated domain.With parallel calculators, this rediscovery of these methods as algorithms of calculations was based on a modern variational approach.Pierre-Louis Lions was the starting point of an intense research activity to develop this tool of calculation, see, e.g., [1,2] and the references therein [3][4][5][6][7][8][9].
In this paper, we give a new approach to the finite element approximation for the problem of variational inequality with noncoercive operator.This problem arises in stochastic control (see [10]).We consider a domain which is the union of two overlapping sub-domains where each sub-domain has its own generated triangulation.To prove the main result of this work, we construct two sequences of subsolutions and we estimate the errors between Schwarz iterates and the subsolutions.The proof stands on a Lipschitz continuous dependency with respect to the source term for variational inequality, while in [5] the proof stands on a Lipschitz continuous dependency with respect to the boundary condition.
The paper is organized as follows.In Sections 2, we introduce the continuous and discrete obstacle problem as well as Schwarz algorithm with two sub-domains and give the geometrical convergence theorem.In Section 3, we establish two sequences of subsolutions and their error estimates and prove a main result concerning the error estimate of solution in the L ∞ -norm, taking into account the combination of geometrical convergence and uniform convergence [11,12] of finite element approximation.

Notations and Assumptions
Let's consider functions ( ) ; , 0 where Ω is a connected bounded domain in 2 R with sufficiently regular boundary ∂Ω .We define a second order differential operator where the bilinear form associated: ( ) , d Let f be a function in ( ) a regular function g defined on ∂Ω such that ( ) AM and a nonempty convex set We assume there exists 0 λ > large enough and a constant 0 then the bilinear form ( ) be the solution of variational inequality (V.I) which is equivalent to .,. denotes the usual inner product in ( ) where ( ) Remark 1.We call quasi-variational inequality (Q.V.I) if the right hand side ( ) depends of solution u , in the contrary case we call variational inequality (V.I).

Some Preliminary Results on the V.I Noncoercive
Thanks to [10], the problem (12) has one and only one solution, moreover u satisfies the regularity property ( )  the correspond- ing solution of V.I (12).
Lemma 1 [10] Under the preceding notations and assumptions (1) to (11), if , that is equivalent to Lemma 2 [10] Under the preceding notations and assumptions (1) to (11), the solution u of problem (12) is the maximum element of the set X .
We show the Lipschitz property, which gives the continuous dependance to the data f .Lemma 3 Under the preceding notations and assumptions (1) to (11), we have where c is an independent constant of data.Proof Firstly, let Secondly, it is clear that by changing the roles of ( ) which completes the proof.Remark 2 If ψ ψ =  and g g =  , then we have Let Ω be decomposed into triangles and let h τ denote the set of those elements; 0 h > is the mesh-size.We assume the triangulation h τ is regular and quasi-uniform.Let h V denote the standard piecewise linear finite element space and by ( )  the basis functions of the space h V .Let h r be the usual restriction operator in Ω .The discrete counterpart of (13) consists of finding where π is an interpolation operator on .

∂Ω
We shall assume that the matrix B defined by is M -matrix [13] (i.e.angles of triangles of h τ are 2 ≤ π ).

The Continuous Schwarz Algorithm
Consider the model obstacle problem: find where ( ) We decompose Ω into two overlapping polygonal subdomains 1 Ω and 2 Ω such that We associate with problem (19) the following system: find ( ) ( ) ( ) Starting from 0 , u ψ = we define the continuous Schwarz sequences ( ) and ( ) Ω such that ( ) 0 in and 0 in .
The following geometrical convergence is due to ( [2], pages 51-63) Theorem 1 The sequences ( ) , 0 n u n + ≥ of the Schwarz algorithm converge geometrically to the solution of the problem (20).More precisely, there exist two constants ] [ such that for all 0 n > ( ) ( )

The Discretization
For 1, 2 i = ; let i h τ be a standard regular and quasi-uniform finite element triangulation in i Ω , i h being the mesh size.We assume that the two triangulations are mutually independent on 1 2 , Ω Ω  where a triangle belonging to one triangulation does not necessarily belong to the other.Let ( ) = Ω be the space of continuous piecewise linear functions on i h τ which vanish on .i ∂Ω ∂Ω  For ( ) Ω such that ( ) and on 2 Ω the sequence ( ) We will also always assume that the respective matrices resulting from problems (23) and ( 24) are M - matrices.

Error Analysis
This section is devoted to the proof of the main result of this work.For that, we begin by introducing two auxiliary sequences.

Auxiliary Schwarz Sequences
To simplify the notation, we take ; is the solution of continuous V.I (21) (resp.( 22)) and let ; is the solution of discrete V.I. (23) (resp.( 24)).It is clear that n ih u is the finite element approximation of .
n i u Then, as , therefore, we apply the error estimate for variational inequality (see [11,12]), we get .

Sequences of Sub-Solutions
The following theorems will play a important role in proving the main result of this paper.

Part One-Discrete Sub-Solution
We construct a discrete function n ih α near n i u such that: .
u + be the solution of (25).Then there exists a function n ih α and a constant c independent of h and , n such that Proof Let us give the proof for 1 i = .The one for 2 i = is similar.Indeed, so, due to lemma 2 (discrete case), it follows that ( ) where and using both remak2 (discrete case) and estimate (27), we get which combined with (29) yields

Part Two-Continuous Sub-Solution
We construct a continuous function ( ), be the solution of (26).Then there exists a function ( ) where ( ), and using both Remark 2 and estimate (28), we get so, combining (31) with estimate (32) yields we get immediately the results.

Conclusion
We have established a convergence order of Schwarz algorithm for two overlapping subdomains with nonmatching grids.This approach developed in this paper relies on the geometrical convergence and the error estimate between the continuous and discrete Schwarz iterates.The constant c in error estimate is independent of Schwarz iterate n.

Γ
We give the discrete counterparts of Schwarz algorithm defined in (21) and (22) as follows.Starting from 0 , ,

Proof
Let us give the proof for 1 i = .The one for 2 i = is similar.indeed, ( ), of V.I (26) for 1 i = , it is also a subsolution, i.e.