Maximum norm analysis of a nonmatching grids method for a class of variational inequalities with nonlinear source terms

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

$$ a(u,v-u)+c(u,v-u)\geq\bigl(f ( u ) ,v-u\bigr),\quad \forall v\in K^{g} , $$

(1.1)

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

$$ c\geq\beta>0, $$

(1.2)

where β is a positive constant, f is a nonlinear Lipschitz functional, $K^{g}$ is a convex set defined by

$$ K^{g}= \bigl\{ v\in H^{1}(\Omega )\text{ such that }v=g\text{ on }\partial \Omega \text{ and }v\leq\psi\text{ on }\Omega \bigr\} $$

(1.3)

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

$$ \bigl\Vert u_{i}-u_{h_{i}}^{n}\bigr\Vert _{L^{\infty} ( \Omega _{i} ) }\leq Ch^{2}\vert \log h\vert ^{2}, $$

(1.4)

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

$$ a ( u,v ) = \int_{\Omega} ( \nabla u\cdot\nabla v ) \,dx, $$

(2.1)

the linear form

$$ ( f,v ) = \int_{\Omega}f(x)\cdot v(x)\,dx, $$

(2.2)

the right-hand side

$$ f\in L^{\infty} ( \Omega ) , $$

(2.3)

the obstacle

$$ \psi\in W^{2,\infty}(\Omega ) \quad \text{such that}\quad \psi\geq0\quad \text{on }\partial \Omega , $$

(2.4)

the boundary condition $g\in L^{\infty} ( \partial\Omega )$, and the nonempty convex set

$$ K^{g}= \bigl\{ v\in H^{1}(\Omega )\text{ such that }v=g\text{ on }\partial \Omega \text{ and }v\leq\psi\text{ on }\Omega \bigr\} . $$

(2.5)

We consider the variational inequality (VI): find $u\in K^{g}$ such that

$$ a(u,v-u)+c(u,v-u)\geq(f,v-u), \quad \forall v\in K^{g}, $$

(2.6)

where $c\in \mathbb{R}$ and $c>0$ such that

$$ c\geq\beta>0, $$

(2.7)

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

$$ a ( u_{h},v-u_{h} ) +c ( u_{h},v-u_{h} ) \geq ( f,v-u_{h} ) , \quad \forall v\in K_{h}^{g} , $$

(2.8)

where

$$ K_{h}^{g}= \{ v\in V_{h}\text{ such that }v= \pi_{h}g\text{ on }\partial\Omega\text{ and }v\leq r_{h}\psi\text{ on }\Omega \} , $$

(2.9)

$\pi_{h}$ 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

$$ \Vert \zeta-\zeta_{h}\Vert _{L^{\infty} ( \Omega ) }\leq Ch^{2} \vert \log h\vert ^{2}. $$

(2.10)

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

$$ a ( \underline{\zeta},v ) +c ( \underline{\zeta},v ) \leq ( f,v ) , \quad \forall v\geq0, $$

with

$$ \underline{\zeta}\leq g\quad \text{on }\partial\Omega. $$

Then the two inequalities $f\leq\tilde{f}$ in Ω and $g\leq \tilde{g}$ on ∂Ω imply

$$ a ( \underline{\zeta},v ) +c ( \underline{\zeta},v ) \leq ( f,v ) \leq ( \tilde{f},v ) ,\quad \forall v\geq0, $$

with

$$ \underline{\zeta}\leq g\leq\tilde{g}\quad \text{on }\partial\Omega. $$

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

$$ \Vert \zeta-\tilde{\zeta} \Vert _{L^{\infty} ( \Omega ) }\leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \Vert f-\tilde{f}\Vert _{L^{\infty} ( \Omega ) }, \Vert g-\tilde{g}\Vert _{L^{\infty} ( \partial\Omega ) } \biggr\} . $$

(2.11)

Proof

First, set

$$ \Phi=\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \Vert f- \tilde{f}\Vert _{L^{\infty} ( \Omega ) },\Vert g-\tilde {g}\Vert _{L^{\infty} ( \partial\Omega ) } \biggr\} . $$

(2.12)

Then

$$\begin{aligned} \tilde{f}& \leq f+\Vert f-\tilde{f}\Vert _{L^{\infty } ( \Omega ) } \\ & \leq f+ ( 1 ) \Vert f-\tilde{f}\Vert _{L^{\infty } ( \Omega ) } \\ & \leq f+ \biggl( \frac{c}{\beta} \biggr) \Vert f-\tilde{f}\Vert _{L^{\infty} ( \Omega ) } \\ & \leq f+c\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \Vert f-\tilde{f}\Vert _{L^{\infty} ( \Omega ) }, \Vert g-\tilde{g}\Vert _{L^{\infty} ( \partial\Omega ) } \biggr\} \\ & \leq f+c\Phi. \end{aligned}$$

So,

$$ \tilde{f}\leq f+c\Phi\quad \text{in }\Omega. $$

(2.13)

On the other hand, we have

$$ \tilde{\zeta}=\tilde{g}\leq g+\Phi\quad \text{on }\partial\Omega. $$

(2.14)

Thus, making use of (2.13), (2.14), and Proposition 2, we obtain

$$ \tilde{\zeta}\leq\sigma ( f+c\Phi ,g+\Phi ) \quad \text{in }\overline{\Omega}. $$

(2.15)

Since $\zeta+\Phi$ is a solution of the following VI:

$$ a \bigl( \zeta+\Phi, ( v+\Phi ) - ( \zeta+\Phi ) \bigr) +c \bigl( \zeta+\Phi, ( v+\Phi ) - ( \zeta+\Phi ) \bigr) \geq \bigl( f+c\Phi, ( v+\Phi ) - ( \zeta +\Phi ) \bigr) $$

with

$$ v+\Phi, \zeta+\Phi\in K^{ ( g+\Phi ) }, $$

we have

$$ \zeta+\Phi=\sigma ( f+c\Phi,g+\Phi ) . $$

(2.16)

Equations (2.15) and (2.16) imply

$$ \tilde{\zeta}\leq\zeta+\Phi\quad \text{in }\overline{\Omega}, $$

thus

$$ \tilde{\zeta}-\zeta\leq\Phi\quad \text{in }\overline{\Omega}. $$

(2.17)

Similarly, interchanging the roles of the couples $(f;g)$; $(\tilde {f},\tilde{g})$, we obtain

$$ \zeta-\tilde{\zeta}\leq\Phi\quad \text{in }\overline{\Omega}, $$

(2.18)

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

$$ \Vert \zeta_{h}-\tilde{\zeta}_{h}\Vert _{L^{\infty } ( \Omega ) } \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \Vert f-\tilde{f}\Vert _{L^{\infty} ( \Omega ) }, \Vert g-\tilde{g}\Vert _{L^{\infty} ( \partial\Omega ) } \biggr\} . $$

(2.19)

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

$$ a ( u,v-u ) +c ( u,v-u ) \geq \bigl( f(u),v-u \bigr), \quad \forall v\in K^{g}, $$

(3.1)

where

$$ a(u,v)= \int_{\Omega} ( \nabla u\nabla v ) \,dx, $$

(3.2)

$f(\cdot)$ is a Lipschitz continuous nondecreasing nonlinear source term on $\mathbb{R}$,

$$ \bigl\vert f ( x ) -f ( y ) \bigr\vert \leq k \vert x-y\vert ,\quad \forall x,y\in \mathbb{R,}$$

(3.3)

with k satisfying

$$ k< \beta, $$

(3.4)

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

$$ \Omega=\Omega_{1}\cup\Omega_{2} . $$

(3.5)

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

$$ \left \{ \textstyle\begin{array}{l} a_{i} ( u_{i},v-u_{i} ) +c ( u_{i},v-u_{i} ) \geq ( f(u_{i}),v-u_{i} ) ,\quad \forall v\in K_{i}^{g} , \\ u_{i}/_{\Gamma_{i}}=u_{j}/_{\Gamma_{i}},\quad i,j=1,2, i\neq j,\end{array}\displaystyle \right . $$

(3.6)

such that

$$\begin{aligned}& K_{i}^{g}= \bigl\{ v\in H^{1}(\Omega _{i})\text{ such that }v=g\text{ on }\partial \Omega \cap\partial \Omega _{i}\text{ and }v\leq\psi\text{ on }\Omega _{i} \bigr\} , \end{aligned}$$

(3.7)

$$\begin{aligned}& a_{i}(u,v)= \int_{\Omega_{i}} ( \nabla u.\nabla v ) \,dx , \end{aligned}$$

(3.8)

and

$$ u_{i}=u/\Omega_{i},\quad i=1,2. $$

(3.9)

Let

$$ b_{i}(u,v)=a_{i}(u,v)+c(u,v). $$

3.1 The continuous Schwarz sequences

Let $u_{2}^{0}$ be an initialization in $\Gamma_{1}$ defined by

$$ u_{2}^{0}=\psi/\Omega_{2}. $$

(3.10)

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

$$ \left \{ \textstyle\begin{array}{l} b_{1} ( u_{1}^{n+1},v-u_{1}^{n+1} ) \geq ( f(u_{1}^{n+1}),v-u_{1}^{n+1} ) ,\quad \forall v\in K_{1}^{g}, \\ u_{1}^{n+1}/_{\Gamma_{1}}=u_{2}^{n}/_{\Gamma1}, \end{array}\displaystyle \right . $$

(3.11)

and $( u_{2}^{n+1} )$ on $\Omega_{2}$ such that $u_{2}^{n+1}\in K_{2}^{g}$ solves

$$ \left \{ \textstyle\begin{array}{l} b_{2} ( u_{2}^{n+1},v-u_{2}^{n+1} ) \geq ( f(u_{2}^{n+1}),v-u_{2}^{n+1} ) , \quad \forall v\in K_{2}^{g}, \\ u_{2}^{n+1}/_{\Gamma_{2}}=u_{1}^{n+1}/_{\Gamma2}. \end{array}\displaystyle \right . $$

(3.12)

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:

$$ V_{h_{i}}= \bigl\{ v\in C ( \overline{\Omega}_{i} ) \cap H^{1} ( \Omega_{i} ) \text{ such that }v/_{T}\in \mathcal{P}_{1}, \forall T\in\tau^{h_{i}} \bigr\} , $$

(3.13)

the convex sets

$$ K_{h_{i}}^{g}= \{ v\in V_{h_{i}}\text{ such that }v=\pi _{h_{i}}g\text{ on }\partial \Omega \cap \partial \Omega _{i}\text{ and }v\leq r_{h_{i}}\psi \}, $$

(3.14)

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

(see [18, 19])

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

$$ \left \{ \textstyle\begin{array}{l} b_{1} ( u_{h_{1}}^{n+1},v-u_{h_{1}}^{n+1} ) \geq ( f(u_{h_{1}}^{n+1}),v-u_{h_{1}}^{n+1} ),\quad \forall v\in K_{h_{1}}^{g}, \\ u_{h_{1}}^{n+1}/_{\Gamma_{1}}=\pi_{h_{1}}(u_{h_{2}}^{n}/_{\Gamma1}),\end{array}\displaystyle \right . $$

(3.15)

and $u_{h_{2}}^{n+1}\in K_{h_{2}}^{g}$ such that

$$ \left \{ \textstyle\begin{array}{l} b_{2} ( u_{h_{2}}^{n+1},v-u_{h_{2}}^{n+1} ) \geq ( f(u_{h_{2}}^{n+1}),v-u_{h_{2}}^{n+1} ),\quad \forall v\in K_{h_{2}}^{g}, \\ u_{h_{2}}^{n+1}/_{\Gamma_{2}}=\pi_{h_{2}}(u_{h_{1}}^{n+1}/_{\Gamma 2}).\end{array}\displaystyle \right . $$

(3.16)

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

$$ \left \{ \textstyle\begin{array}{l} b_{1}(w_{h_{1}},v-w_{h_{1}})\geq ( f ( u_{1} ) ,v-w_{h_{1}} ),\quad \forall v\in K_{h_{1}}^{g}, \\ w_{h_{1}}/_{\Gamma_{1}}=\pi_{h_{1}}(u_{2}/_{\Gamma1}),\end{array}\displaystyle \right . $$

(4.1)

and $w_{h_{2}}\in K_{h_{2}}^{g}$ such that $w_{h_{2}}$ solves

$$ \left \{ \textstyle\begin{array}{l} b_{2}(w_{h_{2}},v-w_{h_{2}})\geq ( f ( u_{2} ) ,v-w_{h_{2}} ),\quad \forall v\in K_{h_{2}}^{g}, \\ w_{h_{2}}/_{\Gamma_{2}}=\pi_{h_{2}}(u_{1}/_{\Gamma2}).\end{array}\displaystyle \right . $$

(4.2)

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

$$ \Vert u_{i}-w_{h_{i}}\Vert _{L^{\infty} ( \Omega _{i} ) }\leq Ch^{2}\vert \log h\vert ^{2}, $$

(4.3)

where C is a constant independent of h.

Notation 8

From now on, we shall adopt the following notations:

$$\begin{aligned}& \vert \cdot \vert _{1} =\Vert \cdot \Vert _{L^{\infty } ( \Gamma_{1} ) };\qquad \vert \cdot \vert _{2}=\Vert \cdot \Vert _{L^{\infty} ( \Gamma_{2} ) }, \\& \Vert \cdot \Vert _{1} =\Vert \cdot \Vert _{L^{\infty } ( \Omega_{1} ) }; \qquad \Vert \cdot \Vert _{2}=\Vert \cdot \Vert _{L^{\infty} ( \Omega_{2} ) }, \\& \pi_{h_{1}} =\pi_{h_{2}}=\pi_{h}. \end{aligned}$$

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

$$ \bigl\Vert u_{i}-u_{h_{i}}^{n+1}\bigr\Vert _{i}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2},\quad i=1,2, n\geq0. $$

(4.4)

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:

$$ (\mathrm{A})\mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq Ch^{2}\vert \log h\vert ^{2} $$

(4.5)

and

$$ (\mathrm{B})\mbox{:}\quad Ch^{2}\vert \log h\vert ^{2}< \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}, $$

(4.6)

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

$$ \Vert \alpha_{h_{i}}-u_{i}\Vert _{i}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \Vert \tilde{\alpha}_{h_{i}}-u_{i}\Vert _{i} \leq\frac {1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

Part 1: The first part of the proof deals with $0<\rho\leq\frac{1}{2}$. So

$$ \frac{\rho}{1-\rho}\leq1. $$

(4.7)

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

$$\begin{aligned}& b_{1} \bigl( w_{h_{1}},\varphi_{s}^{1} \bigr) \leq \bigl( f ( u_{1} ) ,\varphi_{s}^{1} \bigr) , \quad \forall s\in \bigl\{ 1,\ldots,m ( h_{1} ) \bigr\} , \\& w_{h_{1}} =\pi_{h_{1}}u_{2}\quad \text{on } \Gamma_{1}. \end{aligned}$$

Since the nonlinear functional is Lipschitz and according to (4.3), we get

$$ f ( u_{1} ) -f ( w_{h_{1}} ) \leq kCh^{2}\vert \log h\vert ^{2}. $$

Then

$$\begin{aligned}& b_{1} \bigl( w_{h_{1}},\varphi_{s}^{1} \bigr) \leq \bigl( f ( u_{1} ) ,\varphi_{s}^{1} \bigr) \leq \bigl( f ( w_{h_{1}} ) +kCh^{2}\vert \log h\vert ^{2},\varphi_{s}^{1} \bigr), \\& w_{h_{1}} =\pi_{h_{1}}u_{2}\quad \text{on } \Gamma_{1}. \end{aligned}$$

Let

$$ W_{h_{1}}=\sigma_{h_{1}} \bigl( f ( w_{h_{1}} ) +kCh^{2} \vert \log h\vert ^{2},\pi_{h_{1}}u_{2} \bigr) ; $$

(4.8)

therefore, $w_{h_{1}}$ is a subsolution of $W_{h_{1}}$,

$$ w_{h_{1}}\leq W_{h_{1}}\quad \text{in }\Omega_{1}. $$

(4.9)

By applying (2.19), we get

$$\begin{aligned} \begin{aligned} \bigl\Vert W_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}& \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) +kCh^{2}\vert \log h\vert ^{2}-f \bigl( u_{h_{1}}^{1} \bigr) \bigr\Vert _{1} ; \bigl\vert u_{2}-u_{h_{2}}^{0}\bigr\vert _{1} \biggr\} \\ & \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) -f \bigl( u_{h_{1}}^{1} \bigr) \bigr\Vert _{1}+ \biggl( \frac{k}{\beta} \biggr) Ch^{2}\vert \log h \vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \biggr\} . \end{aligned} \end{aligned}$$

So

$$ \bigl\Vert W_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} . $$

(4.10)

On the other hand, (4.9) generates two possibilities, that is,

$$ ( \mathrm{A}_{1} ) \mbox{:}\quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq\bigl\Vert W_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1} $$

or

$$ ( \mathrm{A}_{2} )\mbox{:}\quad \bigl\Vert W_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$

Case (A₁) in conjunction with (4.10) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} , $$

which lets us distinguish the following two cases:

$$\begin{aligned}& 1\mbox{:} \quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} \\& \hphantom{1\mbox{:} \quad}\quad =\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} \end{aligned}$$

(4.11)

and

$$ 2\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} =\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}. $$

(4.12)

Equation (4.11) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}. $$

Then

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$

and

$$\begin{aligned} \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}& \leq\frac{\rho ^{2}}{1-\rho }Ch^{2}\vert \log h\vert ^{2}+\rho Ch^{2}\vert \log h\vert ^{2} \\ & \leq\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq Ch^{2}\vert \log h\vert ^{2}, \end{aligned}$$

which coincides with (4.5) and contradicts (4.6). So, (4.11) is only possible in situation (A). Equation (4.12) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} $$

(4.13)

and

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$

So, by multiplying (4.13) by ρ and adding $\rho Ch^{2} \vert \log h\vert ^{2}$, we get

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h \vert ^{2}. $$

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

$$ ( \mathrm{a} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq \rho \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert $$

or

$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$

That is,

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2} $$

or

$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$

Thus

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}\leq Ch^{2}\vert \log h\vert ^{2} $$

or

$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq Ch^{2}\vert \log h \vert ^{2}. $$

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}. $$

Then (4.12) in situation (A) implies

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2} \vert \log h \vert ^{2}, $$

while in situation (B) only (b) is true and leads to

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\quad \text{and}\quad \frac{\rho}{1-\rho}Ch^{2} \vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}. $$

Then

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} $$

or

$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1} \leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$

We remark that both possibilities are true. There is either a contradiction and (4.12) is impossible or (4.12) is possible only if

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

So, in the two situations (A) and (B) and in the two cases (4.11) and (4.12) of situation (A₁), we get

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} , $$

(4.14)

which implies

$$ w_{h_{1}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}\leq w_{h_{1}}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

Let us denote

$$ \alpha_{h_{1}}=w_{h_{1}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$

(4.15)

and

$$ \tilde{\alpha}_{h_{1}}=w_{h_{1}}+\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2}. $$

(4.16)

Then

$$ \alpha_{h_{1}}\leq u_{h_{1}}^{1}\leq\tilde{ \alpha}_{h_{1}} $$

(4.17)

with

$$\begin{aligned} \Vert \alpha_{h_{1}}-u_{1}\Vert _{1}& =\biggl\Vert w_{h_{1}}- \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}-u_{1}\biggr\Vert _{1} \\ & \leq \Vert w_{h_{1}}-u_{1}\Vert _{1}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} \\ & \leq Ch^{2}\vert \log h\vert ^{2}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} \end{aligned}$$

by virtue of (4.3). So

$$ \Vert \alpha_{h_{1}}-u_{1}\Vert _{1}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.18)

By using the same reasoning we see that (4.16) implies

$$ \Vert \tilde{\alpha}_{h_{1}}-u_{1}\Vert _{1}\leq \frac {1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.19)

On the other hand, (4.17) implies

$$ \alpha_{h_{1}}-u_{1}\leq u_{h_{1}}^{1}-u_{1} \leq\tilde{\alpha}_{h_{1}}-u_{1} $$

(4.20)

so according to (4.18) and (4.19) we get

$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}-u_{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$

(4.21)

thus

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.22)

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

$$ ( \mathrm{c} ) \mbox{:}\quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} $$

or

$$ ( \mathrm{d} ) \mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} \bigr\} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$

(4.23)

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1} $$

and (4.12) in conjunction with (d) implies

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$

Then it is clear that in the two cases (4.11) and (4.12), we obtain

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1} $$

(4.24)

with

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$

(4.25)

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:

$$ ( \mathrm{e} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq Ch^{2}\vert \log h\vert ^{2} $$

or

$$ ( \mathrm{f} ) \mbox{:}\quad \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq Ch^{2}\vert \log h\vert ^{2}. $$

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}, $$

while in situation (B) only the case (f) is true and leads to

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$

In summary, in situation (A₂) and in the two cases (4.11) and (4.12) of situations (A) and (B), we get

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$

(4.26)

Let us decompose the subdomain $\Omega_{1}=\Omega_{1,1}\cup\Omega _{1,1}^{c}$ such that

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\vert \quad \text{on }\Omega _{1,1} $$

(4.27)

and

$$ \bigl\vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\vert < \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\quad \text{on }\Omega_{1,1}^{c}. $$

(4.28)

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,

$$ u_{h_{1}}^{1}-u_{1}\leq\alpha_{h_{1}}-u_{1} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

(4.29)

by virtue of (4.18). On the other hand, (4.18) leads also to

$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq \alpha_{h_{1}}-u_{1}. $$

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:

$$ u_{h_{1}}^{1}-u_{1}\leq-\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$

or

$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}-u_{1}. $$

Then

$$ u_{h_{1}}^{1}-u_{1}\leq-\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2}\leq w_{h_{1}}-u_{1} $$

or

$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{1}-u_{1}\leq w_{h_{1}}-u_{1}. $$

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

$$ \bigl\Vert u_{h_{1}}^{1}-u_{1}\bigr\Vert _{L^{\infty} ( \Omega _{1,1} ) }=\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.30)

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

$$ \bigl\Vert u_{h_{1}}^{1}-u_{1}\bigr\Vert _{L^{\infty} ( \Omega _{1,1}^{c} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.31)

Equations (4.30) and (4.31) imply

$$ \bigl\Vert u_{h_{1}}^{1}-u_{1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.32)

Finally, in the two cases (A₁) and (A₂) and in the two situations (A) and (B), we get

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.33)

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

$$\begin{aligned}& b_{2} \bigl( w_{h_{2}},\varphi_{s}^{2} \bigr) \leq \bigl( f ( u_{2} ) ,\varphi_{s}^{2} \bigr), \quad \forall s\in \bigl\{ 1,\ldots,m ( h_{2} ) \bigr\} , \\& w_{h_{2}} =\pi_{h_{2}}u_{1}\quad \text{on } \Gamma_{2}. \end{aligned}$$

The nonlinear functional is Lipschitz and according to (4.3)

$$ f ( u_{2} ) -f ( w_{h_{2}} ) \leq kCh^{2}\vert \log h\vert ^{2}. $$

Then

$$\begin{aligned}& b_{2} \bigl( w_{h_{2}},\varphi_{s}^{2} \bigr) \leq \bigl( f ( u_{2} ) ,\varphi_{s}^{2} \bigr) \leq \bigl( f ( w_{h_{2}} ) +kCh^{2}\vert \log h\vert ^{2},\varphi_{s}^{2} \bigr) , \\& w_{h_{2}} =\pi_{h_{2}}u_{1}\quad \text{on } \Gamma_{2}. \end{aligned}$$

Let

$$ W_{h_{2}}=\sigma_{h_{2}} \bigl( f ( w_{h_{2}} ) +kCh^{2} \vert \log h\vert ^{2},\pi_{h_{2}}u_{1} \bigr) ; $$

(4.34)

therefore, $w_{h_{2}}$ is a subsolution of $W_{h_{2}}$, so

$$ w_{h_{2}}\leq W_{h_{2}}\quad \text{in }\Omega_{2}. $$

(4.35)

By applying (2.19), we get

$$\begin{aligned} \bigl\Vert W_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}& \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{2}} ) +kCh^{2}\vert \log h\vert -f \bigl( u_{h_{2}}^{1} \bigr) \bigr\Vert _{2} ; \bigl\vert u_{1}-u_{h_{1}}^{1}\bigr\vert _{2} \biggr\} \\ & \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{2}} ) -f \bigl( u_{h_{2}}^{1} \bigr) \bigr\Vert _{2}+ \biggl( \frac{k}{\beta} \biggr) Ch^{2}\vert \log h \vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1} \biggr\} . \end{aligned}$$

So

$$ \bigl\Vert W_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} . $$

(4.36)

On the other hand, (4.35) generates two possibilities, that is,

$$ ( \mathrm{B}_{1} ) \mbox{:}\quad \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}\leq\bigl\Vert W_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2} $$

or

$$ ( \mathrm{B}_{2} ) \mbox{:}\quad \bigl\Vert W_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$

Case (B₁) in conjunction with (4.36) implies that

$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} , $$

which lets us distinguish the following two cases:

$$\begin{aligned}& 1\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} \\& \hphantom{1\mbox{:}\quad}\quad =\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} \end{aligned}$$

(4.37)

and

$$ 2\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} =\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$

(4.38)

Equation (4.37) implies that

$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}. $$

Then

$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$

and

$$\begin{aligned} \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}& \leq\frac{\rho ^{2}}{1-\rho }Ch^{2}\vert \log h\vert ^{2}+\rho Ch^{2}\vert \log h\vert ^{2} \\ & \leq\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2}\vert \log h\vert ^{2}, \end{aligned}$$

which coincides with (4.33). Equation (4.38) implies that

$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1} $$

(4.39)

and

$$ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}. $$

So, by multiplying (4.39) by ρ and adding $\rho Ch^{2} \vert \log h\vert ^{2}$ we get

$$ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h \vert ^{2}; $$

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

$$ ( \mathrm{a} ) \mathrm{:}\quad \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \rho \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert $$

or

$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}, $$

that is,

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2} $$

or

$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \leq\frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2}. $$

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

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2} $$

thus

$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}=\frac{\rho}{1-\rho}Ch^{2} \vert \log h \vert ^{2}. $$

In summary, in the two cases (4.37) and (4.38) of situation (B₁), we get

$$ \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

so

$$ w_{h_{2}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{2}}^{1}\leq w_{h_{2}}+ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

Let us denote

$$ \alpha_{h_{2}}=w_{h_{2}}-\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} $$

(4.40)

and

$$ \tilde{\alpha}_{h_{2}}=w_{h_{2}}+\frac{\rho}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} ; $$

(4.41)

then

$$ \alpha_{h_{2}}\leq u_{h_{2}}^{1}\leq\tilde{ \alpha}_{h_{2}}. $$

(4.42)

By using a same reasoning as adopted in subdomain $\Omega_{1}$ for (4.15) and (4.16), we get

$$ \Vert \alpha_{h_{2}}-u_{2}\Vert _{2}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

(4.43)

and

$$ \Vert \tilde{\alpha}_{h_{2}}-u_{2}\Vert _{2}\leq \frac {1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.44)

Equation (4.42) implies

$$ \alpha_{h_{2}}-u_{2}\leq u_{h_{2}}^{1}-u_{2} \leq\tilde{\alpha}_{h_{2}}-u_{2} $$

and according to (4.43) and (4.44), we obtain

$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{2}}^{1}-u_{2}\leq\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2} , $$

(4.45)

that is,

$$ \bigl\Vert u_{2}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

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,

$$ ( \mathrm{c} ) \mbox{:}\quad \bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}\leq \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} $$

or

$$ ( \mathrm{d} ) \mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \bigr\} \leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$

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

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2} $$

and (4.38) in conjunction with (d) implies

$$ \rho\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1} \leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}. $$

Then the two cases (4.37) and (4.38) imply

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2} $$

and

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$

$\Vert w_{h_{2}}-u_{h_{2}}^{1}\Vert _{2}$ 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:

$$ ( \mathrm{e} ) \mbox{:}\quad \bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}< \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

or

$$ ( \mathrm{f} ) \mbox{:} \quad \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{1}-u_{h_{1}}^{1} \bigr\Vert _{1}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$

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

$$ \bigl\Vert u_{1}-u_{h_{1}}^{1}\bigr\Vert _{1}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1} \bigr\Vert _{2}. $$

So, in the two cases (4.37) and (4.38) of situation (B₂), we get

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{2}}-u_{h_{2}}^{1}\bigr\Vert _{2}. $$

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{1}\bigr\Vert _{L^{\infty} ( \Omega _{2,1} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{1} \bigr\Vert _{L^{\infty} ( \Omega_{2,1}^{c} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$

Finally, in the two situations (B₁) and (B₂) we get

$$ \bigl\Vert u_{2}-u_{h_{2}}^{1}\bigr\Vert _{2}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.46)

Now, let us assume that

$$ \begin{aligned} &\bigl\Vert u_{1}-u_{h_{1}}^{n}\bigr\Vert _{1} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} , \\ &\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} , \end{aligned} $$

(4.47)

and prove that

$$ \begin{aligned} &\bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{1} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert , \\ &\bigl\Vert u_{2}-u_{h_{2}}^{n+1}\bigr\Vert _{2} \leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert . \end{aligned} $$

(4.48)

By using the definition of $W_{h_{1}}$ in (4.8) and by applying (2.19), we get

$$\begin{aligned} \bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}& \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) +kCh^{2}\vert \log h\vert -f \bigl( u_{h_{1}}^{n+1} \bigr) \bigr\Vert _{1} ; \bigl\vert u_{2}-u_{h_{2}}^{n}\bigr\vert _{1} \biggr\} \\ & \leq\max \biggl\{ \biggl( \frac{1}{\beta} \biggr) \bigl\Vert f ( w_{h_{1}} ) -f \bigl( u_{h_{1}}^{n+1} \bigr) \bigr\Vert _{1}+ \biggl( \frac{k}{\beta} \biggr) Ch^{2}\vert \log h \vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2} \biggr\} \end{aligned}$$

so

$$ \bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2} \vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} . $$

(4.49)

On the other hand, (4.9) generates two possibilities, that is

$$ ( \mathrm{C}_{1} ) \mbox{:} \quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}\leq\bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1} $$

or

$$ ( \mathrm{C}_{2} ) \mbox{:} \quad \bigl\Vert W_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}. $$

Case (C₁) in conjunction with (4.49) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\max \bigl\{ \rho \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2} \vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} , $$

which lets us distinguish the following two cases:

$$\begin{aligned} \begin{aligned}[b] &1\mbox{:} \quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} \\ &\hphantom{1\mbox{:} \quad}\quad =\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} \end{aligned} \end{aligned}$$

(4.50)

and

$$ 2\mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} =\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}. $$

(4.51)

Equation (4.50) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}. $$

Then

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho }Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2}, $$

which coincides with (4.47). Equation (4.51) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2} $$

(4.52)

and

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}. $$

So, by multiplying (4.52) by ρ and adding $\rho Ch^{2}\vert \log h\vert ^{2}$ we get

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h \vert ^{2}; $$

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

$$ ( \mathrm{a} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}\leq \rho \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert $$

or

$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}. $$

Thus

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}< \frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2} $$

or

$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \leq\frac{1}{1-\rho}Ch^{2} \vert \log h\vert ^{2}. $$

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}. $$

Then (4.51) implies

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2} \vert \log h \vert ^{2}. $$

Thus, in situation (C₁) and in the two cases (4.50) and (4.51), we get

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

so

$$ \alpha_{h_{1}}\leq u_{h_{1}}^{n+1}\leq\tilde{ \alpha}_{h_{1}} $$

(4.53)

and

$$ \alpha_{h_{1}}-u_{1}\leq u_{h_{1}}^{n+1}-u_{1} \leq\tilde{\alpha}_{h_{1}}-u_{1}. $$

So, according to (4.18) and (4.19), we get

$$ -\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq u_{h_{1}}^{n+1}-u_{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2} \vert \log h\vert ^{2}. $$

Thus

$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

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,

$$ ( \mathrm{c} ) \mbox{:}\quad \bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}\leq \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} $$

or

$$ ( \mathrm{d} ) \mbox{:}\quad \max \bigl\{ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} ; \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \bigr\} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}. $$

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1} $$

and (4.51) in conjunction with (d) implies

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2} \leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}. $$

Then in the two cases (4.50) and (4.51), we get

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}. $$

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:

$$ ( \mathrm{e} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}\leq \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}< \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

or

$$ ( \mathrm{f} ) \mbox{:}\quad \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{n} \bigr\Vert _{2}\leq \frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$

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

$$ \bigl\Vert u_{2}-u_{h_{2}}^{n}\bigr\Vert _{2}=\frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1} \bigr\Vert _{1}; $$

thus, in the two cases (4.50) and (4.51) of situation (C₂), we get

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{n+1}\bigr\Vert _{1}. $$

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

$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{L^{\infty} ( \Omega _{1,1} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1} \bigr\Vert _{L^{\infty} ( \Omega_{1,1}^{c} ) }\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h \vert ^{2}. $$

Finally, in the two situations (C₁) and (C₂) we get the desired result,

$$ \bigl\Vert u_{1}-u_{h_{1}}^{n+1}\bigr\Vert _{1}\leq\frac{1}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

(4.54)

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

$$ \frac{\rho}{1-\rho}>1. $$

(4.55)

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

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}. $$

Then

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$

and

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\frac{\rho}{1-\rho} Ch^{2}\vert \log h\vert ^{2}. $$

So, we can write for (4.5)

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq Ch^{2}\vert \log h\vert ^{2}< \frac{\rho}{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$

and for (4.6)

$$ Ch^{2}\vert \log h\vert ^{2}< \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}. $$

Equation (4.12) implies that

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2} $$

(4.56)

and

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}. $$

So, by multiplying (4.56) by ρ and adding $\rho Ch^{2}\vert \log h\vert ^{2}$ we get

$$ \rho\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}+\rho Ch^{2}\vert \log h\vert ^{2}\leq\rho \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}+\rho Ch^{2}\vert \log h \vert ^{2}. $$

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

$$ ( \mathrm{a} ) \mbox{:}\quad \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq \rho \bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert $$

or

$$ ( \mathrm{b} ) \mbox{:}\quad \rho\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}+\rho Ch^{2}\vert \log h\vert \leq\bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}, $$

that is,

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq Ch^{2}\vert \log h\vert ^{2}< \frac{\rho}{1-\rho}Ch^{2}\vert \log h \vert ^{2} $$

(4.57)

or

$$ \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}\leq \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2} \leq Ch^{2}\vert \log h \vert ^{2}. $$

It is clear that only case (a) is possible because it coincides with (4.5). Equations (4.56) and (4.57) imply

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}\leq\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}, $$

while in (4.6) the two cases (a) and (b) are true with

$$ Ch^{2}\vert \log h\vert ^{2}< \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq \frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2} $$

(4.58)

or

$$ Ch^{2}\vert \log h\vert ^{2}< \frac{\rho}{1-\rho}Ch^{2} \vert \log h\vert ^{2}\leq\bigl\Vert u_{2}-u_{h_{2}}^{0} \bigr\Vert _{2}, $$

which leads to the unique possibility

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}=\frac{\rho}{1-\rho}Ch^{2}\vert \log h\vert ^{2}. $$

In brief, in the two cases (4.11) and (4.12) of situation (A₁) and in the two situations (A) and (B), we get

$$ \bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1}\leq\frac{\rho }{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}. $$

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

$$ \frac{\rho}{ ( 1-\rho ) }Ch^{2}\vert \log h\vert ^{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1}\bigr\Vert _{1} $$

with

$$ \bigl\Vert u_{2}-u_{h_{2}}^{0}\bigr\Vert _{2}\leq\bigl\Vert w_{h_{1}}-u_{h_{1}}^{1} \bigr\Vert _{1}. $$

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.