Abstract
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.
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1 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 [1–4]). The existence, uniqueness, and regularity of the continuous and the discrete solution have been studied and established in the past years (see [3–7]). 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 \(L^{\infty }\)-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 [8–12], and for the nonlinear case we refer to [13–15]. 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 properties 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
2.1 Formulation of the problem
Let Ω be an open bounded polygon in \(R^{2}\) with sufficiently smooth boundary ∂Ω. We define the bilinear form, for any \(u,v\in H^{1}(\Omega)\),
the coefficients \(a_{ij}(x)\), \(a_{j}(x)\), \(a_{0}(x)\) are supposed to be sufficiently smooth and satisfy the following conditions:
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 \(L^{\infty }(\Omega)\) into itself and possesses the following properties [1]:
and a closed convex set
where g is a regular function satisfying
Let \(f(\cdot)\) be the right-hand side supposed nondecreasing and Lipschitz continuous of constant σ such that
We consider the following elliptic quasi-variational inequality (Q.V.I):
\((\cdot,\cdot)\) denotes the usual inner product in \(L^{2}(\Omega)\).
Thanks to [1], the (QVI) (2.11) has a unique solution; moreover, u satisfies the regularity property
Let \(\tau^{h}\) be a standard regular and quasi-uniform finite element triangulation in Ω, h being the mesh size. Let \(V_{h}\) denote the standard piecewise linear finite element space. The discrete counterpart of (2.11) consists of
where
\(r_{h}\) is the usual restriction operator in Ω and \(\pi_{h}\) is an interpolation operator on ∂Ω.
Let \(\varphi_{i}\), \(i=1,2,\ldots,m(h)\), be basis functions of the space \(V_{h}\). We shall assume that the matrix A produced by
is M-matrix [18].
2.2 Monotonicity and \(L^{\infty }\)-stability properties
We consider the linear case, for example, \(f=f(w)\). Let \((f,g)\), \((\widetilde{f},\widetilde{g})\) be a pair of data of linear functions, and
is the solution of inequality
respectively
is the solution of inequality
Then we give the monotonicity result.
Lemma 1
If \(f\geq \widetilde{f}\), \(g\geq \widetilde{g}\), then \(\partial_{h}(f, r_{h}M\xi_{h}, \pi_{h}g)\geq \partial_{h}(\widetilde{f}, r_{h}M \widetilde{\xi }_{h},\pi_{h}\widetilde{g})\).
Proof
Let us reason by recurrence.
For \(n=0\): let \(\xi_{h}^{0}\) (resp. \(\widetilde{\xi }_{h}^{0}\)) be the solution of equation
By the maximum principle, we have
and hence by assumption (2.6)
putting
(resp.)
applying the monotonicity result for (V.I), we get
Now, we define the following sequences:
(resp.)
and we assume that
By (2.6), it follows that
therefore, applying again the monotonicity result for (V.I), we obtain
Finally, if \(n\longrightarrow \infty \) (see [1]), we get
which concludes the proof. □
The proposition below establishes an \(L^{\infty }\)-stability property of the solution with respect to the data.
Proposition 1
Under conditions of Lemma 1, we have
Proof
Firstly, set
we have
and
By summation, we get
and
If we put
then
therefore
where
By (2.7), it follows that
so
Secondly, we have
and
Using Lemma 1, we get
then
Similarly, interchanging the roles of the couples \((f,g) \) and \((\widetilde{f},\widetilde{g})\), we obtain
which completes the proof. □
The following result is due to [6].
Theorem 1
There exists a constant c independent of h such that
2.3 The continuous Schwarz algorithm
We consider the problem: find \(u\in K_{0}(u)\) such that
where \(K_{0}(u)\) is defined in (2.8) with \(g=0\).
We split Ω into two overlapping polygonal sub-domains \(\Omega_{1}\) and \(\Omega_{2}\) such that
and u satisfies the local regularity condition
We set \(\Gamma_{i}=\partial \Omega_{i}\cap \Omega_{j}\), where \(\partial \Omega_{i}\) denotes the boundary of \(\Omega_{i}\). The intersection of \(\Gamma_{1}\) and \(\Gamma_{2}\) is assumed to be empty. We will always assume to simplify that \(\Gamma_{1}\), \(\Gamma_{2}\) are smooth.
For \(w\in C^{0}(\overline{\Gamma }_{i})\), we define
We associate with problem (2.15) the couple \((u_{1},u_{2}) \in V_{1}^{(u_{2})}\times V_{2}^{(u_{1})}\) such that
where
Let \(u^{0}\in C^{0}(\overline{\Omega })\) be the initial value such that
We define the Schwarz sequence \((u_{1}^{n+1})\) on \(\Omega_{1}\) such that \(u_{1}^{n+1}\in V_{1}^{(u_{2}^{n})}\) solves
and respectively \((u_{2}^{n+1})\) on \(\Omega_{2}\) such that \(u_{2}^{n+1} \in V_{2}^{(u_{1}^{n})}\) solves
where
We give a geometrical convergence theorem (see [8]).
Theorem 2
The sequences \((u_{1}^{n+1}, u_{2}^{n+1})\), \(n\geq 0\) converge geometrically to the solution \((u_{1}, u_{2})\) of system (2.16)–(2.17). More precisely, there exist two constants \(0< k_{1},k_{2}<1 \) such that
2.4 The discretization
Let \(\tau^{h_{i}}\) be a standard regular and quasi-uniform finite element triangulation in \(\Omega_{i}\); \(i=1,2\), \(h_{i}\) being the mesh size. We assume that \(\tau^{h_{1}}\) and \(\tau^{h_{2}}\) are mutually independent on \(\Omega_{1}\cap \Omega_{2}\), in the sense that a triangle belonging to \(\tau^{h_{i}}\) does not necessarily belong to \(\tau^{h _{j}}\), \(i\neq j\). Let \(V_{h_{i}}=V_{h_{i}}(\Omega_{i})\) be the space of continuous piecewise linear functions on \(\tau^{h_{i}}\) which vanish on \(\partial \Omega \cap \partial \Omega_{i}\). For given \(w\in C^{0}(\overline{ \Gamma }_{i})\), we set
where \(\pi_{h_{i}}\) denotes a suitable interpolation operator on \(\Gamma_{i}\). We give the discrete counterpart of the Schwarz algorithm defined in (2.19) and (2.20) as follows.
Let \(u_{h_{i}}^{0}=r_{h_{i}}u^{0}\) be given, we define the discrete Schwarz sequence \((u_{1h_{1}}^{n+1})\) on \(\Omega_{1}\) such that \(u_{1h_{1}}^{n+1}\in V_{h_{1}}^{(u_{2h_{2}}^{n})}\) solves
and on \(\Omega_{2}\) the sequence \(u_{2h_{2}}^{n+1}\in V_{h_{2}}^{(u _{1h_{1}}^{n})}\) solves
with
We will also assume that the respective matrices produced by problems (2.21) and (2.22) are M-matrices [18].
3 \(L^{\infty }\)-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.
3.1 Two discrete auxiliary sequences
For \(w_{ih_{i}}^{0}=u_{h_{i}}^{0}\), we define the sequence \(w_{1h_{1}} ^{n+1}\in V_{h_{1}}^{(u_{2}^{n})}\), discrete solution of V.I
respectively the sequence \(w_{2h_{2}}^{n+1}\in V_{h_{2}}^{(u_{1}^{n})}\) satisfies
To simplify the notation, we take
It is clear that \(w_{ih_{i}}^{n}\), \(i=1,2\), is the finite element approximations of \(u_{i}^{n}\) defined in (2.19), (2.20), respectively, where \(f(\cdot)\) is Lipschitz continuous and \(\Vert f(u_{i}^{n}\Vert _{i}\leq c\) (independent of n).
The following lemma will play a crucial role in proving the main result of this paper.
Lemma 2
Let \((u_{i}^{n+1})\), \((u_{ih}^{n+1})\), \(i= 1,2\), be the respective sequences defined in (2.19), (2.20), (2.21), and (2.22). Then there exists a constant c independent of h and n such that
Proof
Let \(\theta =\sigma /\beta \), under assumption (2.10), we have
Let us prove by inductionfor \(n=0\):
Applying Theorem 1 and Proposition 1, putting \(f=f(u_{1}^{0})\), \(\widetilde{f}=f(u_{1h}^{0})\), we obtain
If
then
Making use of an error estimate for elliptic variational equations [19], we obtain
and if
then
Making use again of an error estimate for elliptic variational equations [19], we obtain
Similarly, we have in domain \(\Omega_{2}\)
If
therefore
and if
then
Let us now assume that
and
Consequently,
If
then
and if
therefore
Similarly, we prove the estimate in domain \(\Omega_{2}\). □
3.2 \(L^{\infty }\)-error estimate
Theorem 3
(Main result)
Let \((u_{i}^{n+1})\), \((u_{ih}^{n+1})\), \(i=1,2\), be the respective solutions of (2.19), (2.20), (2.21), and (2.22). Then, for n large enough, there exists a constant c independent of h and n such that
Proof
Let us give the proof for \(i= 1\). The case \(i = 2\) is similar.
Indeed, let \(k=\max (k_{1},k_{2})\). It follows from Theorem 2 and Lemma 2 that
We choose n such that
then
and by inverse inequality, we get
which is the desired error estimate. □
4 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 \(\vert \log h\vert \) 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.
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The authors would like to thank the editors and reviewers for their valuable comments, which greatly improved the readability of this paper. The authors state that no funding source or sponsor has participated in the realization of this work.
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Mehri, A., Saadi, S. A new error estimate on uniform norm of Schwarz algorithm for elliptic quasi-variational inequalities with nonlinear source terms. J Inequal Appl 2018, 60 (2018). https://doi.org/10.1186/s13660-018-1649-3
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DOI: https://doi.org/10.1186/s13660-018-1649-3