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

R E S E A R C H

Open Access

Maximum norm analysis of a

nonmatching grids method for a class of

variational inequalities with nonlinear

source terms

Abida Harbi

*_{Correspondence:}

[email protected]

Laboratory of Applied Mathematics, Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, Annaba, 23000, Algeria

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.

MSC: 65K15; 65N30; 65N15

Keywords: variational inequalities with nonlinear source terms; Schwarz alternating method; nonmatching grids; finite element;L∞-error estimate

1 Introduction

This paper deals with the error analysis in the maximum norm, in the context of the non-matching grids method, of the following variational inequalities with nonlinear source terms: findu∈Kg_{such that}

a(u,v–u) +c(u,v–u)≥f(u),v–u, ∀v∈Kg, (.) wherea(u,v) =(∇u· ∇v)dxis the bilinear form defined in a bounded domainofR orR_{,cis a positive constant such that}

c≥β> , (.)

whereβ is a positive constant,f is a nonlinear Lipschitz functional,Kg is a convex set defined by

Kg=v∈H() such thatv=gon∂andv≤ψon (.) withψ∈W,∞() such thatψ≥ on∂is the obstacle function, andg∈L∞(∂) is the boundary condition.

The concept of the nonmatching finite elements grids used in this paper consists of de-composing the whole domainin two overlapping subdomains and to discretize each subdomain by an independent finite element method. As the two discretizations are in-dependent on the overlap region, the discrete analog of problem (.) 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 [–], and the references therein for the analysis of the Schwarz alternat-ing method for elliptic obstacle problems and to the proceedalternat-ings of the annual domain decomposition conference beginning with [].

For results on maximum norm error analysis of overlapping nonmatching grids methods for elliptic problems we refer, for example, to [–].

In this paper we consider the class of variational inequalities with nonlinear source terms (.) [], 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 [], a Lipschitz continuous depen-dence with respect to both the boundary condition and the source term, and the standard finite elementL∞error estimate for the elliptic obstacle problem [].

More precisely, ifuidenotes the true solution and (un_hi) the discrete Schwarz sequence

with respect to the triangulation with mesh sizehioni, we show that

ui–unhiL∞(i)≤Ch

_|logh|_{, (.)}

whereh=max(h,h),Cis a constant independent ofnandh. This result coincides with the optimal convergence order of elliptic variational inequalities of an obstacle type prob-lem [].

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.

Letbe a bounded polyhedral domain ofRorRwith sufficiently smooth boundary

∂. We consider the bilinear form

a(u,v) =

(∇u· ∇v)dx, (.)

the linear form

(f,v) =

f(x)·v(x)dx, (.)

the right-hand side

f ∈L∞(), (.)

the obstacle

the boundary conditiong∈L∞(∂), and the nonempty convex set

Kg=v∈H() such thatv=gon∂andv≤ψon. (.)

We consider the variational inequality (VI): findu∈Kg_{such that}

a(u,v–u) +c(u,v–u)≥(f,v–u), ∀v∈Kg, (.)

wherec∈Randc>  such that

c≥β> , (.)

whereβ is a positive constant. Letτhbe a triangulation ofwith mesh sizeh,Vhbe the

space of finite elements consisting of continuous piecewise linear functionsvvanishing on∂, andφs,s= , , . . . ,m(h) be the basis functions ofVh.

The discrete counterpart of (.) consists of findinguh∈Khgsuch that

a(uh,v–uh) +c(uh,v–uh)≥(f,v–uh), ∀v∈Khg, (.)

where

K_hg={v∈Vhsuch thatv=πhgon∂andv≤rhψon}, (.)

πhis an interpolation operator on∂andrhis the usual finite element restriction operator

on.

Theorem (see []) Under conditions(.)and(.),there exists a constant C indepen-dent of h such that

ζ–ζhL∞()≤Ch|logh|. (.)

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 (.) with respect to the source term and the boundary condi-tion by which we first set out and demonstrate our result.

Proposition  Let(f;g); (f˜,g)˜ be a pair of data andζ=σ(f,g);ζ˜=σ(f˜,g)˜ the correspond-ing solutions to(.).If f ≤ ˜f inand g≤ ˜g on∂thenζ≤ ˜ζ in.

Proof According to [],ζ=max{ζ}where{ζ}is the set of all the subsolutions ofζ. Hence ∀ζ∈ {ζ},ζ satisfies

with

ζ ≤g on∂.

Then the two inequalitiesf≤ ˜f inandg≤ ˜gon∂imply

a(ζ,v) +c(ζ,v)≤(f,v)≤(f˜,v), ∀v≥,

with

ζ ≤g≤ ˜g on∂.

So,ζis a subsolution ofζ˜=σ(f˜,g), that is,˜ ζ≤ ˜ζ.

Proposition  Under the conditions of Proposition,we have

ζ–ζ˜L∞()≤max 

β

f–f˜L∞(),g–g˜L∞(∂)

. (.)

Proof First, set

=max  β

f–f˜L∞(),g–g˜L∞(∂)

. (.)

Then

˜

f ≤f +f –˜fL∞() ≤f + ()f –f˜L∞() ≤f +

c

β

f –f˜L∞()

≤f +cmax  β

f –f˜L∞(),g–g˜L∞(∂)

≤f +c.

So,

˜

f ≤f +c in. (.)

On the other hand, we have

˜

ζ =g˜≤g+ on∂. (.)

Thus, making use of (.), (.), and Proposition , we obtain

˜

Sinceζ+is a solution of the following VI:

aζ+, (v+) – (ζ+)+cζ+, (v+) – (ζ+)≥f+c, (v+) – (ζ+)

with

v+,ζ+∈K(g+), we have

ζ+=σ(f+c,g+). (.)

Equations (.) and (.) imply

˜

ζ ≤ζ+ in,

thus

˜

ζ–ζ≤ in. (.)

Similarly, interchanging the roles of the couples (f;g); (f˜,g), we obtain˜

ζ–ζ˜≤ in, (.)

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 anM-matrix (see [, ]), 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  Let(f,g); (f˜,g)˜ be a pair of data andζh=σh(f,g);ζ˜h=σh(f˜,g)˜ the

corre-sponding solutions to(.).If f ≥ ˜f inand g≥ ˜g on∂thenζh≥ ˜ζhin.

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  Provided that the d.m.p.is verified,then under the conditions of Proposi-tion,we have

ζh–ζ˜hL∞()≤max 

β

f–f˜L∞(),g–g˜L∞(∂)

. (.)

Proof The proof is similar to that of the continuous case. Indeed, as the basis functionsφs

3 Schwarz alternating methods for VI with nonlinear source terms

We consider the following variational inequality with nonlinear source term (.): Find u∈Kg_{, a solution of}

a(u,v–u) +c(u,v–u)≥f(u),v–u, ∀v∈Kg, (.) where

a(u,v) =

(∇u∇v)dx, (.)

f(·) is a Lipschitz continuous nondecreasing nonlinear source term onR,

f(x) –f(y)≤k|x–y|, ∀x,y∈R, (.)

withksatisfying

k<β, (.)

whereβis defined in (.).

Theorem (see []) Problem(.)has a unique solution.

We decomposeinto two overlapping smooth subdomainsandsuch that

=∪. (.)

We denote by∂ithe boundary ofiandi=∂i∩j. We assume that the intersection

ofiandj,i=j, is empty and we associate with problem (.) the following system: Find

(u,u)∈Kg×K

g

, a solution of

ai(ui,v–ui) +c(ui,v–ui)≥(f(ui),v–ui), ∀v∈Kig,

ui/i=uj/i, i,j= , ,i=j,

(.)

such that

K_ig=v∈H(i) such thatv=gon∂∩∂iandv≤ψoni

, (.)

ai(u,v) =

i

(∇u.∇v)dx, (.)

and

ui=u/i, i= , . (.)

Let

3.1 The continuous Schwarz sequences

Letu

be an initialization indefined by

u_=ψ/. (.)

We, respectively, define the alternating Schwarz sequences (un+

 ) onsuch thatun+∈ K_g solves

b(un+,v–un+)≥(f(un+),v–un+), ∀v∈K

g

, un_+/=u

n

/,

(.)

and (un+) onsuch thatun+∈K

g

solves

b(un+,v–un+)≥(f(un+),v–un+), ∀v∈K

g

, un+

 /=un+/.

(.)

Theorem (see []) The two sequences(.)and(.)converge uniformly to the solution of(.).

3.2 Nonmatching grids discretization

Fori= , , letτhi_{be a standard regular and quasi-uniform finite element triangulation} ini;hibeing its mesh size. The two meshes being mutually independent on∩in the sense that a triangle belonging to one triangulation does not necessarily belong to the other one. We consider the following discrete spaces:

Vhi=

v∈C(i)∩H(i) such thatv/T∈P,∀T∈τhi

, (.)

the convex sets

K_hg_i={v∈Vhisuch thatv=πhigon∂∩∂iandv≤rhiψ}, (.) whererhidenotes the restriction operator on the triangulationτ

hi_{. Let alsoπ}

hidenote the interpolation operator oniandφis,s= , , . . . ,m(hi), be the basis functions ofVhi. The discrete maximum principle(see [, ]) We assume that the respective matrices resulting from the discretization of problems (.) and (.) are M-matrices. Note that, as the two mesheshandhare independent over the overlapping subdomains, it is im-possible to formulate a global approximate problem which would be the direct discrete counterpart of problem (.).

3.3 The discrete Schwarz sequences

Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (.) and (.). Indeed, letu

h be the discrete analog ofu



 defined in (.) that is, u

h=πh(u) =πh(ψ/). We, respectively, defineunh+ ∈K

g

hsuch that

b(unh+ ,v–u

n+

h )≥(f(u

n+

h ),v–u

n+

h ), ∀v∈K

g h,

un+

h /=πh(u

n h/),

andun_h+_ ∈K_hg_ such that

b(unh+ ,v–u

n+

h )≥(f(u

n+

h ),v–u

n+

h ), ∀v∈K

g h,

un_h+_ /=πh(u

n+

h /).

(.)

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 definewh∈K

g

hsuch thatwhsolves

b(wh,v–wh)≥(f(u),v–wh), ∀v∈K

g h,

wh/=πh(u/),

(.)

andwh∈K

g

h such thatwh solves

b(wh,v–wh)≥(f(u),v–wh), ∀v∈K

g h,

wh/ =πh(u/).

(.)

It is then clear thatwhandwhare the finite element approximations ofuandudefined

in (.) thus, making use of (.), we get

ui–whiL∞(i)≤Ch

_|logh|_{, (.)}

whereCis a constant independent ofh.

Notation  From now on, we shall adopt the following notations:

| · |= · L∞(); | · |= · L∞(),

· = · L∞(); · = · L∞(), πh=πh=πh.

4.2 The main result

Theorem  Let h=max(h,h)and letρ=_βk < then there exists a constant C indepen-dent of both h and n such that

ui–unh+i i≤

 ( –ρ)Ch

_|logh|_{, i= , ,n≥. (.)}

Proof The proof of (.) will be carried out by induction, where the casesρ∈(,_] and

ρ∈(_, ) will be studied separately. Also, within each case, we will also discuss the two following situations:

(A): u–uh≤Ch

and

(B): Ch|logh|<u–uh, (.) whereu

h=πh(u



) =πh(ψ/). The basic idea of the proof is to define for each subdo-main two approximationsαhiandα˜hiin theL∞-norm ofui(a discrete subsolution and a discrete supersolution ofun

hi,n≥), such that αhi–uii≤

 ( –ρ)Ch

_|logh|

and

˜αhi–uii≤  ( –ρ)Ch

_|logh|_.

Part: The first part of the proof deals with  <ρ≤_. So

ρ

 –ρ ≤. (.)

Forn= , in domain , the discrete analogwhofudefined in (.) considered as the upper

bound of the set of discrete subsolutions [], satisfies

b

wh,ϕ



s

≤f(u),ϕs

, ∀s∈, . . . ,m(h)

,

wh=πhu on.

Since the nonlinear functional is Lipschitz and according to (.), we get

f(u) –f(wh)≤kCh

_|logh|_.

Then

b

wh,ϕ



s

≤f(u),ϕs

≤f(wh) +kCh

_|logh|_,ϕ

s

,

wh=πhu on.

Let

Wh=σh

f(wh) +kCh

_|logh|_,π

hu

; (.)

therefore,whis a subsolution ofWh,

wh≤Wh in. (.)

By applying (.), we get

Wh–u



h≤max 

β

f(wh) +kCh

_|logh|_–fu

h;u–u 

h

≤max  β

f(wh) –f

u_h+

k

β

Ch|logh|;u–uh

So

Wh–u



h≤max

ρwh–u



h+ρCh

_|logh|_;u

–uh

. (.)

On the other hand, (.) generates two possibilities, that is,

(A): wh–u



h≤Wh–u



h or

(A): Wh–u



h≤wh–u



h. Case (A) in conjunction with (.) implies that

wh–u



h≤max

ρwh–u



h+ρCh

_|logh|_;u

–uh

,

which lets us distinguish the following two cases:

: maxρwh–u



h+ρCh

_|logh|_;u

–uh

=ρwh–u



h+ρCh

_|logh| _(.)

and

: maxρwh–u



h+ρCh

_|logh|_;u

–uh

=u–uh. (.) Equation (.) implies that

wh–u



h≤ρwh–u



h+ρCh _|logh|

and

u–uh≤ρwh–u



h+ρCh

_|logh|_.

Then

wh–u



h≤

ρ

 –ρCh

_|logh|

and

u–uh≤

ρ  –ρCh

_|logh|_+ρCh_|logh|

≤ ρ  –ρCh

_|logh|_≤Ch_|logh|_,

which coincides with (.) and contradicts (.). So, (.) is only possible in situation (A). Equation (.) implies that

wh–u



h≤u–u 

and

ρwh–u



h+ρCh

_|logh|_≤u

–uh.

So, by multiplying (.) byρand addingρCh_|logh|_{, we get}

ρwh–u



h+ρCh

_|logh|_≤ρu

–uh+ρCh

_|logh|_.

Then ρwh–uh+ρCh

_|logh| _{is bounded by bothu}

–uh andρu–u



h+ ρCh|logh|, so

(a): u–uh≤ρu–u 

h+ρCh _|logh|

or

(b): ρu–uh+ρCh

_{|logh| ≤u}

–uh. That is,

_u–u_h≤ ρ  –ρCh

_|logh|

or

ρ

 –ρCh

_|logh|_≤u

–uh. Thus

u–uh≤

ρ

 –ρCh

_|logh|_≤Ch_|logh|

or

ρ

 –ρCh

_|logh|_≤u

–uh≤Ch

_|logh|_.

So, the two cases (a) and (b) are true because they both coincide with (.). Therefore, there is either a contradiction and thus (.) is impossible or (.) is possible only if

u–uh=

ρ

 –ρCh

_|logh|_.

Then (.) in situation (A) implies

wh–u



h≤u–u 

h=

ρ

 –ρCh

_|logh|_,

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

wh–u



h≤u–u 

h and

ρ

 –ρCh

_|logh|_≤u

Then

wh–u



h≤

ρ

 –ρCh

_|logh|_≤u

–uh or

ρ

 –ρCh

_|logh|_≤w

h–u



h≤u–u 

h.

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

wh–u



h=

ρ

( –ρ)Ch

_|logh|_.

So, in the two situations (A) and (B) and in the two cases (.) and (.) of situation (A), we get

wh–u



h≤

ρ

( –ρ)Ch

_|logh|_{, (.)}

which implies

wh– ρ

( –ρ)Ch

_|logh|_≤u

h≤wh+ ρ

( –ρ)Ch

_|logh|_.

Let us denote

αh=wh– ρ

( –ρ)Ch

_|logh| _(.)

and

˜

αh=wh+ ρ

( –ρ)Ch

_|logh|_{. (.)}

Then

αh≤u



h≤ ˜αh (.)

with

αh–u= wh–

ρ

( –ρ)Ch

_|logh|_–u 



≤ wh–u+ ρ

( –ρ)Ch _|logh|

≤Ch|logh|+ ρ ( –ρ)Ch

_|logh|

by virtue of (.). So

αh–u≤

 ( –ρ)Ch

By using the same reasoning we see that (.) implies

˜αh–u≤

 ( –ρ)Ch

_|logh|_{. (.)}

On the other hand, (.) implies

αh–u≤uh–u≤ ˜αh–u (.)

so according to (.) and (.) we get

–  ( –ρ)Ch

_|logh|_≤u

h–u≤

 ( –ρ)Ch

_|logh| _(.)

thus

u–uh≤  ( –ρ)Ch

_|logh|_{. (.)}

Case (A) in conjunction with (.) implies thatWh–uh is bounded by the values

wh–u



handmax{ρwh–u



h+ρCh

_|logh|_;u

–uh}which generates the two

situations

(c): wh–u



h≤max

ρwh–u



h+ρCh

_|logh|_;u

–uh

or

(d): maxρwh–u



h+ρCh

_|logh|_;u

–uh

≤wh–u



h. (.) It is clear that case (c) coincides with situation (A). Let us study case (d); as in case (A),

max{ρwh–u



h+ρCh

_|logh|_;u

–uh}lets us distinguish the two cases (.) and

(.). Equation (.) in conjunction with (d) implies

u–uh≤ρwh–u



h+ρCh

_|logh|_≤w

h–u



h and (.) in conjunction with (d) implies

ρwh–u



h+ρCh

_|logh|_≤u

–uh≤wh–u



h. Then it is clear that in the two cases (.) and (.), we obtain

ρ

( –ρ)Ch

_|logh|_≤w

h–u



h (.)

with

u–uh≤wh–u



Thus,wh–uhis bounded below by both ρ

(–ρ)Ch|logh|andu–u 

hso we

dis-tinguish the two following possibilities:

(e): u–uh≤

ρ

( –ρ)Ch

_|logh|_≤Ch_|logh|

or

(f): ρ ( –ρ)Ch

_|logh|_≤u

–uh≤Ch

_|logh|_.

So, the two cases (e) and (f ) are true because they both coincide with (.). Therefore, there is either a contradiction and thus cases (.) and (.) are impossible or the two cases (.) and (.) are possible in situation (A) only if

u–uh=

ρ

( –ρ)Ch

_|logh|_≤w

h–u



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

ρ

( –ρ)Ch

_|logh|_≤u

–uh≤wh–u



h.

In summary, in situation (A) and in the two cases (.) and (.) of situations (A) and (B), we get

ρ

( –ρ)Ch

_|logh|_≤w

h–u



h. (.)

Let us decompose the subdomain=,∪c,such that

ρ

( –ρ)Ch

_|logh|_≤w

h–u



h on, (.)

and

wh–u



h< ρ

( –ρ)Ch

_|logh| _onc

,. (.)

We begin with,. Ifwh–uh≥ on,then (.) impliesu



h≤αh; thus,

u_h–u≤αh–u≤

 ( –ρ)Ch

_|logh| _(.)

by virtue of (.). On the other hand, (.) leads also to

–  ( –ρ)Ch

_|logh|_≤α

h–u.

So,αh–uis bounded below by bothu



h–uand –

 (–ρ)Ch

_|logh|_{, which lets us}

distin-guish the two following possibilities:

u_h–u≤–  ( –ρ)Ch

or

–  ( –ρ)Ch

_|logh|_≤u

h–u.

Then

u_h–u≤–  ( –ρ)Ch

_|logh|_≤w

h–u

or

–  ( –ρ)Ch

_|logh|_≤u

h–u≤wh–u.

So, both possibilities are true because they coincide with (.). So, there is either a con-tradiction and (.) is impossible or (.) is possible and we must have

u_h–u_L∞₍,)=

 ( –ρ)Ch

_|logh|_{. (.)}

The casewh–uh<  on,is studied in a similar manner and leads to the same result

(.). Equation (.) is studied in the same way as that for case (A) and leads to

u_h–u_L∞₍c

,)≤

 ( –ρ)Ch

_|logh|_{. (.)}

Equations (.) and (.) imply

u

h–u≤  ( –ρ)Ch

_|logh|_{. (.)}

Finally, in the two cases (A) and (A) and in the two situations (A) and (B), we get

u–uh≤  ( –ρ)Ch

_|logh|_{. (.)}

Forn=  in domain , the discrete analogwhofu, defined in (.) and considered as the

upper bound of the set of discrete subsolutions [], satisfies

b

wh,ϕ



s

≤f(u),ϕs

, ∀s∈, . . . ,m(h)

,

wh=πhu on.

The nonlinear functional is Lipschitz and according to (.)

f(u) –f(wh)≤kCh

_|logh|_.

Then

b

wh,ϕ



s

≤f(u),ϕs

≤f(wh) +kCh

_|logh|_,ϕ

s

,

Let

Wh=σh

f(wh) +kCh

_|logh|_,π

hu

; (.)

therefore,wh is a subsolution ofWh, so

wh≤Wh in. (.)

By applying (.), we get

Wh–u



h≤max 

β

f(wh) +kCh

_|logh|–fu

h;u–u 

h

≤max  β

f(wh) –f

u_h+

k

β

Ch|logh|;u–uh

.

So

Wh–u



h≤max

ρwh–u



h+ρCh

_|logh|_;u –uh

. (.)

On the other hand, (.) generates two possibilities, that is,

(B): wh–u



h≤Wh–u



h or

(B): Wh–u



h≤wh–u



h. Case (B) in conjunction with (.) implies that

wh–u



h≤max

ρwh–u



h+ρCh

_|logh|_;u –uh

,

which lets us distinguish the following two cases:

: maxρwh–u



h+ρCh

_|logh|_;u –uh

=ρwh–u



h+ρCh

_|logh| _(.)

and

: maxρwh–u



h+ρCh

_|logh|_;u

–uh

=u–uh. (.) Equation (.) implies that

wh–u



h≤ρwh–u



h+ρCh _|logh|

and

u–uh≤ρwh–u



h+ρCh

Then

wh–u



h≤

ρ

 –ρCh

_|logh|

and

u–uh≤

ρ

 –ρCh

_|logh|_+ρCh_|logh|

≤ ρ  –ρCh

_|logh|_<   –ρCh

_|logh|_,

which coincides with (.). Equation (.) implies that

wh–u



h≤u–u 

h (.)

and

ρwh–u



h+ρCh

_|logh|_≤u

–uh.

So, by multiplying (.) byρand addingρCh_|logh|_{we get}

ρwh–u



h+ρCh

_|logh|_≤ρu

–uh+ρCh

_|logh|_;

thenρwh –uh+ρCh

_|logh| _{is bounded above byu}

–u_h andρu–u_h+

ρCh_{|logh|. So, either}

(a): u–uh≤ρu–u 

h+ρCh _|logh|

or

(b): ρu–uh+ρCh

_{|logh| ≤u}

–uh, that is,

u–uh≤

ρ

 –ρCh

_|logh|_<   –ρCh

_|logh|

or

ρ

 –ρCh

_|logh|_≤u

–uh≤   –ρCh

_|logh|_.

So, the two cases (a) and (b) are true because they both coincide with (.). Therefore, there is either a contradiction and thus (.) is impossible or (.) is possible only if

u–uh=

ρ

 –ρCh

_|logh|

thus

wh–u



h≤u–u 

h=

ρ

 –ρCh

In summary, in the two cases (.) and (.) of situation (B), we get

_wh –u



h≤

ρ

( –ρ)Ch _|logh|

so

wh– ρ

( –ρ)Ch

_|logh|_≤u

h≤wh+ ρ

( –ρ)Ch

_|logh|_.

Let us denote

αh=wh– ρ

( –ρ)Ch

_|logh| _(.)

and

˜

αh=wh+ ρ

( –ρ)Ch

_|logh|_{; (.)}

then

αh≤u



h≤ ˜αh. (.)

By using a same reasoning as adopted in subdomainfor (.) and (.), we get

αh–u≤

 ( –ρ)Ch

_|logh| _(.)

and

˜αh–u≤

 ( –ρ)Ch

_|logh|_{. (.)}

Equation (.) implies

αh–u≤u



h–u≤ ˜αh–u

and according to (.) and (.), we obtain

–  ( –ρ)Ch

_|logh|_≤u

h–u≤

 ( –ρ)Ch

_|logh|_{, (.)}

that is,

u–uh≤  ( –ρ)Ch

_|logh|_.

Case (B) in conjunction with (.) implies thatWh–uhis bounded by the values

wh –uh andmax{ρwh –uh+ρCh

_|logh|_;u

 –u_h}, which generates two situations,

(c): wh–u



h≤max

ρwh–u



h+ρCh

_|logh|_;u

–uh

or

(d): maxρwh–u



h+ρCh

_|logh|_;u

–uh

≤wh–u



h.

It is clear that case (c) coincides with case (B). Let us study case (d); as in case (B)

max{ρwh –uh+ρCh

_|logh|_;u

–u_h}lets us distinguish the two cases (.) and (.). Equation (.) in conjunction with (d) implies

u–uh≤ρwh–u



h+ρCh

_|logh|_≤w

h–u



h and (.) in conjunction with (d) implies

ρwh–u



h+ρCh

_|logh|_≤u

–uh≤wh–u



h. Then the two cases (.) and (.) imply

ρ

( –ρ)Ch

_|logh|_≤w

h–u



h and

u–uh≤wh–u



h. wh–uhis bounded below by

ρ

(–ρ)Ch

_|logh|_andu

–uh so we distinguish the

two following possibilities:

(e): u–uh≤

ρ

( –ρ)Ch

_|logh|_<  ( –ρ)Ch

_|logh|

or

(f): ρ ( –ρ)Ch

_|logh|_≤u

–uh≤  ( –ρ)Ch

_|logh|_.

So, the two cases (e) and (f ) are true because they both coincide with (.). Therefore, there is either a contradiction and thus cases (.) and (.) are impossible or the two cases (.) and (.) are possible only if

u–uh=

ρ

( –ρ)Ch

_|logh|_≤w

h–u



h. So, in the two cases (.) and (.) of situation (B), we get

ρ

( –ρ)Ch

_|logh|_≤w

h–u



h.

The remainder of the proof related to situation (B) rests on the same arguments used in subdomain for situation (A) that is, on a decomposition of=,∪c,and showing that

u–uhL∞(,)≤

 ( –ρ)Ch

and

u–uhL∞(c_,)≤  ( –ρ)Ch

_|logh|_.

Finally, in the two situations (B) and (B) we get

u–uh≤  ( –ρ)Ch

_|logh|_{. (.)}

Now, let us assume that

u–unh≤  ( –ρ)Ch

_|logh|_,

u–unh≤  ( –ρ)Ch

_|logh|_,

(.)

and prove that

_u–un_h+_{ }≤  ( –ρ)Ch

_|logh|,

u–unh+ ≤  ( –ρ)Ch

_|logh|.

(.)

By using the definition ofWhin (.) and by applying (.), we get Wh–u

n+

h ≤max 

β

f(wh) +kCh

_|logh|–fun+

h ;u–u

n h

≤max  β

f(wh) –f

un_h+_{ }+

k

β

Ch|logh|;u–unh

so

Wh–u

n+

h ≤max

ρwh–u

n+

h +ρCh

_|logh|_;u

–unh

. (.)

On the other hand, (.) generates two possibilities, that is

(C): wh–u

n+

h ≤Wh–u

n+

h  or

(C): Wh–u

n+

h ≤wh–u

n+

h . Case (C) in conjunction with (.) implies that

wh–u

n+

h ≤max

ρwh–u

n+

h +ρCh

_|logh|_;u

–unh

,

which lets us distinguish the following two cases:

: maxρwh–u

n+

h +ρCh

_|logh|_;u

–unh

=ρwh–u

n+

h +ρCh

and

: maxρwh–u

n+

h +ρCh

_|logh|_;u

–unh

=u–unh. (.)

Equation (.) implies that

wh–u

n+

h ≤ρwh–u

n+

h +ρCh _|logh|

and

u–unh≤ρwh–u

n+

h +ρCh

_|logh|_.

Then

wh–u

n+

h ≤

ρ

 –ρCh

_|logh|

and

u–unh≤

ρ

 –ρCh

_|logh|_<   –ρCh

_|logh|_,

which coincides with (.). Equation (.) implies that

wh–u

n+

h ≤u–u

n

h (.)

and

ρwh–u

n+

h +ρCh

_|logh|_≤u

–unh.

So, by multiplying (.) byρand addingρCh_|logh|_{we get}

ρwh–u

n+

h +ρCh

_|logh|_≤ρu

–unh+ρCh

_|logh|_;

thenρwh–u

n+

h +ρCh

_|logh| _{is bounded by bothu}

–unh andρu–u

n h+ ρCh_{|logh|. So}

(a): u–unh≤ρu–u

n

h+ρCh _|logh|

or

(b): ρu–unh+ρCh

_{|logh| ≤u}

–unh. Thus

u–unh≤

ρ

 –ρCh

_|logh|_<   –ρCh

or

ρ

 –ρCh

_|logh|_≤u

–unh≤   –ρCh

_|logh|_.

So, the two cases (a) and (b) are true because they both coincide with (.). Therefore, there is either a contradiction and thus (.) is impossible or (.) is possible only if

u–unh=

ρ

 –ρCh

_|logh|_.

Then (.) implies

wh–u

n+

h ≤u–u

n h=

ρ

 –ρCh

_|logh|_.

Thus, in situation (C) and in the two cases (.) and (.), we get

wh–u

n+

h ≤

ρ

( –ρ)Ch _|logh|

so

αh≤u

n+

h ≤ ˜αh (.)

and

αh–u≤u

n+

h –u≤ ˜αh–u.

So, according to (.) and (.), we get

–  ( –ρ)Ch

_|logh|_≤un+

h –u≤

 ( –ρ)Ch

_|logh|_.

Thus

u–unh+ ≤  ( –ρ)Ch

_|logh|_.

Case (C) in conjunction with (.) implies thatWh–unh+ is bounded by the values

wh–unh+andmax{ρwh–u

n+

h +ρCh

_|logh|_;u

–un_h}, which generates two situations,

(c): wh–u

n+

h ≤max

ρwh–u

n+

h +ρCh

_|logh|_;u

–unh

or

(d): maxρwh–u

n+

h +ρCh

_|logh|_;u

–unh

≤wh–u

n+

h .

It is clear that case (c) coincides with case (C). Let us study case (d); as in case (C),

max{ρwh–u

n+

h +ρCh

_|logh|_;u

and (.). Equation (.) in conjunction with (d) implies

u–unh≤ρwh–u

n+

h +ρCh

_|logh|_≤w

h–u

n+

h  and (.) in conjunction with (d) implies

ρwh–u

n+

h +ρCh

_|logh|_≤u

–unh≤wh–u

n+

h . Then in the two cases (.) and (.), we get

ρ

( –ρ)Ch

_|logh|_≤w

h–u

n+

h  and

u–unh≤wh–u

n+

h .

Hence,wh–unh+  is bounded below by both ρ

(–ρ)Ch

_|logh|_andu

–unh so we

distinguish the two following possibilities:

(e): u–unh≤

ρ

( –ρ)Ch

_|logh|_<  ( –ρ)Ch

_|logh|

or

(f): ρ ( –ρ)Ch

_|logh|_≤u

–unh≤  ( –ρ)Ch

_|logh|_.

So, the two cases (e) and (f ) are true because they both coincide with (.). Therefore, there is either a contradiction and the two cases (.) and (.) are impossible or the two cases (.) and (.) are possible only if

u–unh=

ρ

( –ρ)Ch

_|logh|_≤w

h–u

n+

h ;

thus, in the two cases (.) and (.) of situation (C), we get

ρ

( –ρ)Ch

_|logh|_≤w

h–u

n+

h .

The remainder of the proof related to situation (C) rests on the same arguments used in subdomainfor situation (A) at iterationn= , that is, on a decomposition of=

,∪c,and on showing that

u–uhn+L∞(,)≤

 ( –ρ)Ch

_|logh|

and

u–unh+ L∞(c_,)≤  ( –ρ)Ch

Finally, in the two situations (C) and (C) we get the desired result,

_u–un+

h ≤  ( –ρ)Ch

_|logh|_{. (.)}

Estimate (.) in domain  can be proved similarly using estimate (.). Part: This second part of the proof is devoted to_<ρ< . So

ρ

 –ρ > . (.)

Forn= , in domain , like as in part , (.) generates two different situations (A) and (A), which we study separately. According to (.), situation (A) in conjunction with (.) implies

wh–u



h≤ρwh–u



h+ρCh _|logh|

and

u–uh≤ρwh–u



h+ρCh

_|logh|_.

Then

wh–u



h≤

ρ

 –ρCh

_|logh|

and

u–uh≤

ρ

 –ρCh

_|logh|_.

So, we can write for (.)

_u–u_h≤Ch|logh|< ρ  –ρCh

_|logh|

and for (.)

Ch|logh|<u–uh≤

ρ

 –ρCh

_|logh|_.

Equation (.) implies that

wh–u



h≤u–u 

h (.)

and

ρwh–u



h+ρCh

_|logh|_≤u

–uh.

So, by multiplying (.) byρand addingρCh|logh|we get

ρwh–u



h+ρCh

_|logh|_≤ρu

–uh+ρCh