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Maximum norm analysis of a nonmatching grids method for a class of variational inequalities with nonlinear source terms
Journal of Inequalities and Applications volume 2016, Article number: 181 (2016)
Abstract
In this paper, we study a nonmatching grid finite element approximation of a class of elliptic variational inequalities with nonlinear source terms in the context of the Schwarz alternating domain decomposition. We show that the approximation converges optimally in the maximum norm, on each subdomain, making use of a Lipschitz continuous dependence with respect to both the boundary condition and the source term.
1 Introduction
This paper deals with the error analysis in the maximum norm, in the context of the nonmatching grids method, of the following variational inequalities with nonlinear source terms: find \(u\in K^{g}\) such that
where \(a(u,v)=\int_{\Omega} ( \nabla u\cdot\nabla v ) \,dx\) is the bilinear form defined in a bounded domain Ω of \(\mathbb{R} ^{2}\) or \(\mathbb{R} ^{3}\), c is a positive constant such that
where β is a positive constant, f is a nonlinear Lipschitz functional, \(K^{g}\) is a convex set defined by
with \(\psi\in W^{2,\infty}(\Omega )\) such that \(\psi\geq0\) on ∂Ω is the obstacle function, and \(g\in L^{\infty} ( \partial\Omega ) \) is the boundary condition.
The concept of the nonmatching finite elements grids used in this paper consists of decomposing the whole domain Ω in two overlapping subdomains and to discretize each subdomain by an independent finite element method. As the two discretizations are independent on the overlap region, the discrete analog of problem (1.1) cannot be defined, and the alternating Schwarz method is then used to resolve the two discrete subproblems arising from these nonmatching finite elements grids.
We refer to [1–7], and the references therein for the analysis of the Schwarz alternating method for elliptic obstacle problems and to the proceedings of the annual domain decomposition conference beginning with [8].
For results on maximum norm error analysis of overlapping nonmatching grids methods for elliptic problems we refer, for example, to [9–14].
In this paper we consider the class of variational inequalities with nonlinear source terms (1.1) [15], where the main objective is to demonstrate that the approximation converges optimally on each subdomain making use of the characterization of the discrete solution as the upper bound of the set of discrete subsolutions [16], a Lipschitz continuous dependence with respect to both the boundary condition and the source term, and the standard finite element \(L^{\infty}\) error estimate for the elliptic obstacle problem [17].
More precisely, if \(u_{i}\) denotes the true solution and \(( u_{h_{i}}^{n} ) \) the discrete Schwarz sequence with respect to the triangulation with mesh size \(h_{i}\) on \(\Omega_{i}\), we show that
where \(h=\max ( h_{1},h_{2} ) \), C is a constant independent of n and h. This result coincides with the optimal convergence order of elliptic variational inequalities of an obstacle type problem [17].
2 Elliptic variational inequalities of obstacle type problem
In this section we begin by laying down some definitions and classical results related to variational inequalities, then we prove a Lipschitz continuous and discrete dependence with respect to both the boundary condition and the source term which will assume a crucial role in the proof of the main result of this paper.
Let Ω be a bounded polyhedral domain of \(\mathbb{R} ^{2}\) or \(\mathbb{R} ^{3}\) with sufficiently smooth boundary ∂Ω. We consider the bilinear form
the linear form
the right-hand side
the obstacle
the boundary condition \(g\in L^{\infty} ( \partial\Omega )\), and the nonempty convex set
We consider the variational inequality (VI): find \(u\in K^{g}\) such that
where \(c\in \mathbb{R} \) and \(c>0\) such that
where β is a positive constant. Let \(\tau_{h}\) be a triangulation of Ω with mesh size h, \(V_{h}\) be the space of finite elements consisting of continuous piecewise linear functions v vanishing on ∂Ω, and \(\phi_{s}\), \(s=1,2,\ldots,m(h)\) be the basis functions of \(V_{h}\).
The discrete counterpart of (2.6) consists of finding \(u_{h}\in K_{h}^{g}\) such that
where
\(\pi_{h}\) is an interpolation operator on ∂Ω and \(r_{h}\) is the usual finite element restriction operator on Ω.
Theorem 1
(see [17])
Under conditions (2.3) and (2.4), there exists a constant C independent of h such that
2.1 A Lipschitz continuous dependence with respect to both the boundary condition and the source term
This subsection is devoted to the establishment of a Lipschitz continuous dependence property of the solution with respect to the data whose proof is based on a monotonicity property of the solution of (2.6) with respect to the source term and the boundary condition by which we first set out and demonstrate our result.
Proposition 2
Let \((f;g)\); \((\tilde{f},\tilde{g})\) be a pair of data and \(\zeta =\sigma(f,g)\); \(\tilde{\zeta}=\sigma(\tilde{f},\tilde{g})\) the corresponding solutions to (2.6). If \(f\leq \tilde{f}\) in Ω and \(g\leq\tilde{g}\) on ∂Ω then \(\zeta \leq\tilde{\zeta}\) in Ω̅.
Proof
According to [16], \(\zeta=\max \{ \underline{\zeta} \} \) where \(\{ \underline{\zeta} \} \) is the set of all the subsolutions of ζ. Hence \(\forall \underline{\zeta}\in \{ \underline{\zeta} \} \), \(\underline{\zeta}\) satisfies
with
Then the two inequalities \(f\leq\tilde{f}\) in Ω and \(g\leq \tilde{g}\) on ∂Ω imply
with
So, \(\underline{\zeta}\) is a subsolution of \(\tilde {\zeta}=\sigma ( \tilde{f},\tilde{g} ) \), that is, \(\zeta\leq \tilde{\zeta}\). □
Proposition 3
Under the conditions of Proposition 2, we have
Proof
First, set
Then
So,
On the other hand, we have
Thus, making use of (2.13), (2.14), and Proposition 2, we obtain
Since \(\zeta+\Phi\) is a solution of the following VI:
with
we have
Equations (2.15) and (2.16) imply
thus
Similarly, interchanging the roles of the couples \((f;g)\); \((\tilde {f},\tilde{g})\), we obtain
which completes the proof. □
2.2 A Lipschitz discrete dependence with respect to both the boundary condition and the source term
Assuming that the discrete maximum principle (d.m.p.) is satisfied, i.e. the matrix resulting from the finite element discretization is an M-matrix (see [18, 19]), we prove the Lipschitz discrete dependence with respect to both the boundary condition and the source term by a similar study to that undertaken previously for the Lipschitz continuous dependence property.
Proposition 4
Let \((f,g)\); \((\tilde{f},\tilde{g})\) be a pair of data and \(\zeta_{h}=\sigma_{h}(f,g)\); \(\tilde{\zeta}_{h}=\sigma_{h}(\tilde {f},\tilde{g})\) the corresponding solutions to (2.8). If \(f\geq\tilde{f}\) in Ω and \(g\geq\tilde{g}\) on ∂Ω then \(\zeta_{h}\geq\tilde{\zeta}_{h}\) in Ω̅.
Proof
The proof is similar to that of the continuous case. □
The proposition below establishes a Lipschitz discrete dependence of the solution with respect to the data.
Proposition 5
Provided that the d.m.p. is verified, then under the conditions of Proposition 4, we have
Proof
The proof is similar to that of the continuous case. Indeed, as the basis functions \(\phi_{s}\) of the space \(V_{h}\) are positive, it suffices to use the discrete maximum principle. □
3 Schwarz alternating methods for VI with nonlinear source terms
We consider the following variational inequality with nonlinear source term (1.1): Find \(u\in K^{g}\), a solution of
where
\(f(\cdot)\) is a Lipschitz continuous nondecreasing nonlinear source term on \(\mathbb{R} \),
with k satisfying
where β is defined in (1.2).
Theorem 6
(see [20])
Problem (3.1) has a unique solution.
We decompose Ω into two overlapping smooth subdomains \(\Omega_{1}\) and \(\Omega_{2}\) such that
We denote by \(\partial\Omega_{i}\) the boundary of \(\Omega_{i}\) and \(\Gamma_{i}=\partial\Omega_{i}\cap\Omega_{j}\). We assume that the intersection of \(\overline{\Gamma}_{i}\) and \(\overline{\Gamma}_{j}\), \(i\neq j\), is empty and we associate with problem (3.1) the following system: Find \((u_{1},u_{2})\in K_{1}^{g}\times K_{2}^{g}\), a solution of
such that
and
Let
3.1 The continuous Schwarz sequences
Let \(u_{2}^{0}\) be an initialization in \(\Gamma_{1}\) defined by
We, respectively, define the alternating Schwarz sequences \(( u_{1}^{n+1} ) \) on \(\Omega_{1}\) such that \(u_{1}^{n+1}\in K_{1}^{g}\) solves
and \(( u_{2}^{n+1} ) \) on \(\Omega_{2}\) such that \(u_{2}^{n+1}\in K_{2}^{g}\) solves
Theorem 7
(see [6])
The two sequences (3.11) and (3.12) converge uniformly to the solution of (3.6).
3.2 Nonmatching grids discretization
For \(i=1,2\), let \(\tau^{h_{i}}\) be a standard regular and quasi-uniform finite element triangulation in \(\Omega_{i}\); \(h_{i}\) being its mesh size. The two meshes being mutually independent on \(\Omega_{1}\cap\Omega_{2}\) in the sense that a triangle belonging to one triangulation does not necessarily belong to the other one. We consider the following discrete spaces:
the convex sets
where \(r_{h_{i}}\) denotes the restriction operator on the triangulation \(\tau^{h_{i}}\). Let also \(\pi_{h_{i}}\) denote the interpolation operator on \(\Gamma_{i}\) and \(\phi_{s}^{i}\), \(s=1,2,\ldots,m(h_{i})\), be the basis functions of \(V_{h_{i}}\).
The discrete maximum principle
We assume that the respective matrices resulting from the discretization of problems (3.11) and (3.12) are M-matrices. Note that, as the two meshes \(h_{1}\) and \(h_{2}\) are independent over the overlapping subdomains, it is impossible to formulate a global approximate problem which would be the direct discrete counterpart of problem (3.1).
3.3 The discrete Schwarz sequences
Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (3.11) and (3.12). Indeed, let \(u_{h_{2}}^{0}\) be the discrete analog of \(u_{2}^{0}\) defined in (3.10) that is, \(u_{h_{2}}^{0}=\pi_{h_{2}} ( u_{2}^{0} ) =\pi_{h_{2}} ( \psi /\Omega_{2} ) \). We, respectively, define \(u_{h_{1}}^{n+1}\in K_{h_{1}}^{g}\) such that
and \(u_{h_{2}}^{n+1}\in K_{h_{2}}^{g}\) such that
4 Maximum norm error
This section is devoted to the proof of the main result of the present paper. To that end, we begin by introducing two discrete auxiliary problems.
4.1 Two auxiliary problems
We define \(w_{h_{1}}\in K_{h_{1}}^{g}\) such that \(w_{h_{1}}\) solves
and \(w_{h_{2}}\in K_{h_{2}}^{g}\) such that \(w_{h_{2}}\) solves
It is then clear that \(w_{h_{1}}\) and \(w_{h_{2}}\) are the finite element approximations of \(u_{1}\) and \(u_{2}\) defined in (3.6) thus, making use of (2.10), we get
where C is a constant independent of h.
Notation 8
From now on, we shall adopt the following notations:
4.2 The main result
Theorem 9
Let \(h=\max ( h_{1},h_{2} ) \) and let \(\rho=\frac{k}{\beta}<1\) then there exists a constant C independent of both h and n such that
Proof
The proof of (4.4) will be carried out by induction, where the cases \(\rho \in (0,\frac{1}{2}]\) and \(\rho\in(\frac{1}{2},1)\) will be studied separately. Also, within each case, we will also discuss the two following situations:
and
where \(u_{h_{2}}^{0}=\pi_{h_{2}} ( u_{2}^{0} ) =\pi _{h_{2}} ( \psi/\Omega_{2} ) \). The basic idea of the proof is to define for each subdomain two approximations \(\alpha_{h_{i}}\) and \(\tilde {\alpha}_{h_{i}}\) in the \(L^{\infty}\)-norm of \(u_{i}\) (a discrete subsolution and a discrete supersolution of \(u_{h_{i}}^{n}\), \(n\geq1 \)), such that
and
Part 1: The first part of the proof deals with \(0<\rho\leq\frac{1}{2}\). So
For \(n=1\), in domain 1, the discrete analog \(w_{h_{1}}\) of \(u_{1} \) defined in (4.1) considered as the upper bound of the set of discrete subsolutions [16], satisfies
Since the nonlinear functional is Lipschitz and according to (4.3), we get
Then
Let
therefore, \(w_{h_{1}}\) is a subsolution of \(W_{h_{1}}\),
By applying (2.19), we get
So
On the other hand, (4.9) generates two possibilities, that is,
or
Case (A1) in conjunction with (4.10) implies that
which lets us distinguish the following two cases:
and
Equation (4.11) implies that
and
Then
and
which coincides with (4.5) and contradicts (4.6). So, (4.11) is only possible in situation (A). Equation (4.12) implies that
and
So, by multiplying (4.13) by ρ and adding \(\rho Ch^{2} \vert \log h\vert ^{2}\), we get
Then \(\rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded by both \(\Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}\) and \(\rho \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}+\rho Ch^{2}\vert \log h \vert \), so
or
That is,
or
Thus
or
So, the two cases (a) and (b) are true because they both coincide with (4.5). Therefore, there is either a contradiction and thus (4.12) is impossible or (4.12) is possible only if
Then (4.12) in situation (A) implies
while in situation (B) only (b) is true and leads to
Then
or
We remark that both possibilities are true. There is either a contradiction and (4.12) is impossible or (4.12) is possible only if
So, in the two situations (A) and (B) and in the two cases (4.11) and (4.12) of situation (A1), we get
which implies
Let us denote
and
Then
with
by virtue of (4.3). So
By using the same reasoning we see that (4.16) implies
On the other hand, (4.17) implies
so according to (4.18) and (4.19) we get
thus
Case (A2) in conjunction with (4.10) implies that \(\Vert W_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}\) is bounded by the values \(\Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}\) and \(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2} \} \) which generates the two situations
or
It is clear that case (c) coincides with situation (A1). Let us study case (d); as in case (A1), \(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2} \} \) lets us distinguish the two cases (4.11) and (4.12). Equation (4.11) in conjunction with (d) implies
and (4.12) in conjunction with (d) implies
Then it is clear that in the two cases (4.11) and (4.12), we obtain
with
Thus, \(\Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}\) is bounded below by both \(\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\) and \(\Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}\) so we distinguish the two following possibilities:
or
So, the two cases (e) and (f) are true because they both coincide with (4.5). Therefore, there is either a contradiction and thus cases (4.11) and (4.12) are impossible or the two cases (4.11) and (4.12) are possible in situation (A) only if
while in situation (B) only the case (f) is true and leads to
In summary, in situation (A2) and in the two cases (4.11) and (4.12) of situations (A) and (B), we get
Let us decompose the subdomain \(\Omega_{1}=\Omega_{1,1}\cup\Omega _{1,1}^{c}\) such that
and
We begin with \(\Omega_{1,1}\). If \(w_{h_{1}}-u_{h_{1}}^{1}\geq0\) on \(\Omega_{1,1}\) then (4.27) implies \(u_{h_{1}}^{1}\leq \alpha_{h_{1}}\); thus,
by virtue of (4.18). On the other hand, (4.18) leads also to
So, \(\alpha_{h_{1}}-u_{1}\) is bounded below by both \(u_{h_{1}}^{1}-u_{1}\) and \(-\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\), which lets us distinguish the two following possibilities:
or
Then
or
So, both possibilities are true because they coincide with (4.3). So, there is either a contradiction and (4.27) is impossible or (4.27) is possible and we must have
The case \(w_{h_{1}}-u_{h_{1}}^{1}<0\) on \(\Omega_{1,1}\) is studied in a similar manner and leads to the same result (4.30). Equation (4.28) is studied in the same way as that for case (A1) and leads to
Equations (4.30) and (4.31) imply
Finally, in the two cases (A1) and (A2) and in the two situations (A) and (B), we get
For \(n=1\) in domain 2, the discrete analog \(w_{h_{2}}\) of \(u_{2}\), defined in (4.2) and considered as the upper bound of the set of discrete subsolutions [16], satisfies
The nonlinear functional is Lipschitz and according to (4.3)
Then
Let
therefore, \(w_{h_{2}}\) is a subsolution of \(W_{h_{2}}\), so
By applying (2.19), we get
So
On the other hand, (4.35) generates two possibilities, that is,
or
Case (B1) in conjunction with (4.36) implies that
which lets us distinguish the following two cases:
and
Equation (4.37) implies that
and
Then
and
which coincides with (4.33). Equation (4.38) implies that
and
So, by multiplying (4.39) by ρ and adding \(\rho Ch^{2} \vert \log h\vert ^{2}\) we get
then \(\rho \Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded above by \(\Vert u_{1}-u_{h_{1}}^{1}\Vert _{1}\) and \(\rho \Vert u_{1}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h \vert \). So, either
or
that is,
or
So, the two cases (a) and (b) are true because they both coincide with (4.33). Therefore, there is either a contradiction and thus (4.38) is impossible or (4.38) is possible only if
thus
In summary, in the two cases (4.37) and (4.38) of situation (B1), we get
so
Let us denote
and
then
By using a same reasoning as adopted in subdomain \(\Omega_{1}\) for (4.15) and (4.16), we get
and
Equation (4.42) implies
and according to (4.43) and (4.44), we obtain
that is,
Case (B2) in conjunction with (4.36) implies that \(\Vert W_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}\) is bounded by the values \(\Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}\) and \(\max \{ \rho \Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{1}-u_{h_{1}}^{1}\Vert _{1} \} \), which generates two situations,
or
It is clear that case (c) coincides with case (B1). Let us study case (d); as in case (B1) \(\max \{ \rho \Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{1}-u_{h_{1}}^{1}\Vert _{1} \} \) lets us distinguish the two cases (4.37) and (4.38). Equation (4.37) in conjunction with (d) implies
and (4.38) in conjunction with (d) implies
Then the two cases (4.37) and (4.38) imply
and
\(\Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}\) is bounded below by \(\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\) and \(\Vert u_{1}-u_{h_{1}}^{1}\Vert _{1}\) so we distinguish the two following possibilities:
or
So, the two cases (e) and (f) are true because they both coincide with (4.33). Therefore, there is either a contradiction and thus cases (4.37) and (4.38) are impossible or the two cases (4.37) and (4.38) are possible only if
So, in the two cases (4.37) and (4.38) of situation (B2), we get
The remainder of the proof related to situation (B2) rests on the same arguments used in subdomain \(\Omega_{1}\) for situation (A2) that is, on a decomposition of \(\Omega _{2}=\Omega _{2,1}\cup\Omega_{2,1}^{c}\) and showing that
and
Finally, in the two situations (B1) and (B2) we get
Now, let us assume that
and prove that
By using the definition of \(W_{h_{1}}\) in (4.8) and by applying (2.19), we get
so
On the other hand, (4.9) generates two possibilities, that is
or
Case (C1) in conjunction with (4.49) implies that
which lets us distinguish the following two cases:
and
Equation (4.50) implies that
and
Then
and
which coincides with (4.47). Equation (4.51) implies that
and
So, by multiplying (4.52) by ρ and adding \(\rho Ch^{2}\vert \log h\vert ^{2}\) we get
then \(\rho \Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded by both \(\Vert u_{2}-u_{h_{2}}^{n}\Vert _{2}\) and \(\rho \Vert u_{2}-u_{h_{2}}^{n}\Vert _{2}+\rho Ch^{2}\vert \log h \vert \). So
or
Thus
or
So, the two cases (a) and (b) are true because they both coincide with (4.47). Therefore, there is either a contradiction and thus (4.51) is impossible or (4.51) is possible only if
Then (4.51) implies
Thus, in situation (C1) and in the two cases (4.50) and (4.51), we get
so
and
So, according to (4.18) and (4.19), we get
Thus
Case (C2) in conjunction with (4.52) implies that \(\Vert W_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1} \) is bounded by the values \(\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}\) and \(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}+\rho Ch^{2} \vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{n}\Vert _{2} \} \), which generates two situations,
or
It is clear that case (c) coincides with case (C1). Let us study case (d); as in case (C1), \(\max \{ \rho \Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \Vert u_{2}-u_{h_{2}}^{n}\Vert _{2} \} \) lets us distinguish the two cases (4.50) and (4.51). Equation (4.50) in conjunction with (d) implies
and (4.51) in conjunction with (d) implies
Then in the two cases (4.50) and (4.51), we get
and
Hence, \(\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\Vert _{1}\) is bounded below by both \(\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\) and \(\Vert u_{2}-u_{h_{2}}^{n}\Vert _{2}\) so we distinguish the two following possibilities:
or
So, the two cases (e) and (f) are true because they both coincide with (4.47). Therefore, there is either a contradiction and the two cases (4.50) and (4.51) are impossible or the two cases (4.50) and (4.51) are possible only if
thus, in the two cases (4.50) and (4.51) of situation (C2), we get
The remainder of the proof related to situation (C2) rests on the same arguments used in subdomain \(\Omega_{1}\) for situation (A2) at iteration \(n=1\), that is, on a decomposition of \(\Omega_{1}=\Omega_{1,1}\cup\Omega_{1,1}^{c}\) and on showing that
and
Finally, in the two situations (C1) and (C2) we get the desired result,
Estimate (4.48) in domain 2 can be proved similarly using estimate (4.54).
Part 2: This second part of the proof is devoted to \(\frac{1}{2}<\rho<1\). So
For \(n=1\), in domain 1, like as in part 1, (4.9) generates two different situations (A1) and (A2), which we study separately. According to (4.10), situation (A1) in conjunction with (4.11) implies
and
Then
and
So, we can write for (4.5)
and for (4.6)
Equation (4.12) implies that
and
So, by multiplying (4.56) by ρ and adding \(\rho Ch^{2}\vert \log h\vert ^{2}\) we get
Then \(\rho \Vert w_{h_{1}}-u_{h_{1}}^{1}\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\) is bounded by \(\Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}\) and \(\rho \Vert u_{2}-u_{h_{2}}^{0}\Vert _{2}+\rho Ch^{2}\vert \log h \vert \), so
or
that is,
or
It is clear that only case (a) is possible because it coincides with (4.5). Equations (4.56) and (4.57) imply
while in (4.6) the two cases (a) and (b) are true with
or
which leads to the unique possibility
In brief, in the two cases (4.11) and (4.12) of situation (A1) and in the two situations (A) and (B), we get
The rest of the proof is similar to the part 1, situation (A1), and leads to the result (4.33). According to (4.10), situation (A2) like in part 1 focuses on the study of the case (d) and in the two cases (4.11) and (4.12), we get
with
The rest of the proof related to situation (A2) is similar to part 1, situation (A2), and leads to the result (4.33). That is, in the two situations (A) and (B) with \(\frac{1}{2}<\rho<1\), we get (4.33).
The remainder of the proof related to \(\frac{1}{2}<\rho<1\) is by induction and similar to part 1, by which we obtain the desired result (4.4). □
5 Conclusion
In this paper an optimal convergence order for finite element Schwarz alternating method for a class of VI with nonlinear source terms on two subdomains with nonmatching grids is obtained. The approach rests on a discrete Lipschitz dependence with respect to the both boundary condition and the source term. This approach offers practical perspectives in that it enables us to control the error, on each subdomain between the discrete Schwarz algorithm and the true solution.
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Harbi, A. Maximum norm analysis of a nonmatching grids method for a class of variational inequalities with nonlinear source terms. J Inequal Appl 2016, 181 (2016). https://doi.org/10.1186/s13660-016-1110-4
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DOI: https://doi.org/10.1186/s13660-016-1110-4