A new error estimate on uniform norm of Schwarz algorithm for elliptic quasi-variational inequalities with nonlinear source terms

The Schwarz algorithm for a class of elliptic quasi-variational inequalities with nonlinear source terms is studied in this work. The authors prove a new error estimate in uniform norm, making use of a stability property of the discrete solution. The domain is split into two sub-domains with overlapping non-matching grids. This approach combines the geometrical convergence of solutions and the uniform convergence of variational inequalities.


Introduction
In the present paper, we consider the numerical solution of elliptic quasi-variational inequalities with nonlinear right-hand side. This kind of problem has many applications in impulse control (see [1][2][3][4]). The existence, uniqueness, and regularity of the continuous and the discrete solution have been studied and established in the past years (see [3][4][5][6][7]). To estimate a new error of the solution, we apply the Schwarz algorithm, so we split the domain into two overlapping sub-domains such that each sub-domain has its own generated triangulations. In this approach we transform the nonlinear problem into a sequence of linear problems in each sub-domain.
To prove the main result of this paper, we construct two discrete auxiliary sequences of Schwarz, and we estimate the error between continuous and discrete Schwarz iterates. The proof is based on a discrete L ∞ -stability property with respect to both the boundary condition and the source term for variational inequality, while in [8] the proof is based on a stability property with respect to the boundary condition for variational inequality. Regarding research in this domain, for the linear case we refer the reader to [8][9][10][11][12], and for the nonlinear case we refer to [13][14][15]. The analysis of geometrical convergence of the Schwarz algorithm has been proven in [8,16,17].
This paper consists of two parts. In the first, we formulate the problem of continuous and discrete quasi-variational inequality, we show the monotonicity and stability proper-ties of discrete solution, then we define the Schwarz algorithm for two sub-domains with overlapping non-matching grids. In the second part, we establish two auxiliary Schwarz sequences, and we prove the main result of this work.
2 An overlapping Schwarz method for elliptic quasi-variational inequalities with nonlinear source terms

Formulation of the problem
Let be an open bounded polygon in R 2 with sufficiently smooth boundary ∂ . We define the bilinear form, for any u, v ∈ H 1 ( ), the coefficients a ij (x), a j (x), a 0 (x) are supposed to be sufficiently smooth and satisfy the following conditions: 1≤i,j≤2 We also suppose that the bilinear form is continuous and strongly coercive Let the obstacle Mu of impulse control be defined by The operator M maps L ∞ ( ) into itself and possesses the following properties [1]: and a closed convex set where g is a regular function satisfying Let f (·) be the right-hand side supposed nondecreasing and Lipschitz continuous of constant σ such that σ /β < 1. (2.10) We consider the following elliptic quasi-variational inequality (Q.V.I): (2.11) (·, ·) denotes the usual inner product in L 2 ( ). Thanks to [1], the (QVI) (2.11) has a unique solution; moreover, u satisfies the regularity property Let τ h be a standard regular and quasi-uniform finite element triangulation in , h being the mesh size. Let V h denote the standard piecewise linear finite element space. The discrete counterpart of (2.11) consists of

13)
r h is the usual restriction operator in and π h is an interpolation operator on ∂ . Let ϕ i , i = 1, 2, . . . , m(h), be basis functions of the space V h . We shall assume that the matrix A produced by is M-matrix [18].

Monotonicity and L ∞ -stability properties
We consider the linear case, for example, f = f (w). Let (f , g), ( f , g) be a pair of data of linear functions, and is the solution of inequality Then we give the monotonicity result.
Proof Let us reason by recurrence.
By the maximum principle, we have and hence by assumption (2.6) applying the monotonicity result for (V.I), we get Now, we define the following sequences: and we assume that By (2.6), it follows that therefore, applying again the monotonicity result for (V.I), we obtain Finally, if n − → ∞ (see [1]), we get which concludes the proof.
The proposition below establishes an L ∞ -stability property of the solution with respect to the data.

Proposition 1 Under conditions of Lemma 1, we have
Proof Firstly, set we have By summation, we get If we put By (2.7), it follows that Secondly, we have Using Lemma 1, we get Similarly, interchanging the roles of the couples (f , g) and ( f , g), we obtain which completes the proof.
The following result is due to [6].

The continuous Schwarz algorithm
We consider the problem: find u ∈ K 0 (u) such that We split into two overlapping polygonal sub-domains 1 and 2 such that and u satisfies the local regularity condition We set i = ∂ i ∩ j , where ∂ i denotes the boundary of i . The intersection of 1 and 2 is assumed to be empty. We will always assume to simplify that 1 , 2 are smooth.
For w ∈ C 0 ( i ), we define We associate with problem (2.15) the couple (u 1 , Let u 0 ∈ C 0 ( ) be the initial value such that We define the Schwarz sequence (u n+1 1 ) on 1 such that u n+1 and respectively (u n+1 2 ) on 2 such that u n+1 where u 0 1 = u 0 in 1 , u 0 2 = u 0 in 2 , We give a geometrical convergence theorem (see [8]).

The discretization
Let τ h i be a standard regular and quasi-uniform finite element triangulation in i ; i = 1, 2, h i being the mesh size. We assume that τ h 1 and τ h 2 are mutually independent on 1 ∩ 2 , in the sense that a triangle belonging to τ h i does not necessarily belong to where π h i denotes a suitable interpolation operator on i . We give the discrete counterpart of the Schwarz algorithm defined in (2.19) and (2.20) as follows. Let u 0 h i = r h i u 0 be given, we define the discrete Schwarz sequence (u n+1 1h 1 ) on 1 such that u n+1 and on 2 the sequence u n+1 We will also assume that the respective matrices produced by problems (2.21) and (2.22) are M-matrices [18].

L ∞ -error analysis
The aim of this section is to show the main result of this paper. To that end, we start by introducing two discrete auxiliary sequences and prove a fundamental lemma.

Two discrete auxiliary sequences
respectively the sequence w n+1 (3.2) To simplify the notation, we take It is clear that w n ih i , i = 1, 2, is the finite element approximations of u n i defined in (2.19), (2.20), respectively, where f (·) is Lipschitz continuous and f (u n i i ≤ c (independent of n). The following lemma will play a crucial role in proving the main result of this paper. Proof Let θ = σ /β, under assumption (2.10), we have θ < 1.
Let us prove by inductionfor n = 0: Applying Theorem 1 and Proposition 1, putting f = f (u 0 1 ), f = f (u 0 1h ), we obtain Making use of an error estimate for elliptic variational equations [19], we obtain and if Making use again of an error estimate for elliptic variational equations [19], we obtain Similarly, we have in domain 2 Let us now assume that u n 1u n 1h 1 ≤ cnh 2 | log h| 2 and u n 2u n 2h 2 ≤ cnh 2 | log h| 2 . Consequently, If max θ u n 1u n 1h 1 , u n 2u n 2h 1 = θ u n 1u n 1h 1 , then u n+1 1u n+1 1h 1 ≤ ch 2 | log h| 2 + θ u n 1u n Similarly, we prove the estimate in domain 2 .

L ∞ -error estimate
Proof Let us give the proof for i = 1. The case i = 2 is similar.

Conclusion
In this work, we have established a new approach of an overlapping Schwarz algorithm on non-matching grids for a class of elliptic quasi-variational inequalities with nonlinear source terms. We have obtained a new error estimate in uniform norm which is optimal for these problems. The error estimate obtained contains a logarithmic factor with an extra power of | log h| than expected. We will see that this result plays an important role in the study of an error estimate for evolutionary problems with nonlinear source terms.