Approximation of the two-dimensional Dirichlet problem by one-dimensional continuous and discrete problems on one-dimensional networks

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Approximation of the two-dimensional Dirichlet problem by one-dimensional continuous and discrete problems on

one-dimensional networks

Maryse Bourlard-Jospin, Serge Nicaise, Juliette Venel

To cite this version:

Maryse Bourlard-Jospin, Serge Nicaise, Juliette Venel. Approximation of the two-dimensional Dirich- let problem by one-dimensional continuous and discrete problems on one-dimensional networks. 2013.

�hal-00834307�

Approximation of the two-dimensional Dirichlet problem by one-dimensional continuous and discrete problems on

one-dimensional networks

Maryse Bourlard-Jospin, Serge Nicaise and Juliette Venel

^∗

March 29, 2013

Abstract

We show that the solution of the two-dimensional Dirichlet problem set in a plane do- main is the limit of the solutions of similar problems set on a sequence of one-dimensional networks as their size goes to zero. Roughly speaking this means that a membrane can be seen as the limit of rackets made of strings. For practical applications, we also show that the solutions of the discrete approximated problems (again on the one-dimensional networks) also converge to the solution of the two-dimensional Dirichlet problem.

MOS subjet classification: 35R02, 35B40, 65N30

1 Introduction

Approximation of multidimensional boundary value problems by discrete problems or by bound- ary value problems set on less dimensional ones is very important in practice. For discrete ap- proximations, the most popular methods are the finite difference method or the finite element method, for which a lot of convergence results are proved [2, 21]. By the less dimensional approx- imation, we mean that a n-dimensional problem is approximated by a family of k-dimensional ones with k < n. For instance the approximation of boundary value problems set on objects of R

³

with a small thickness by boundary value problems set on objects of dimension 1 or 2 was largely considered in the literature, see for instance [5, 3, 18]. In the same spirit, let us also mention homogenization techniques that analyze the limit process of problems set on n-dimensional domains of thickness to problems still set on domains of dimension n [4].

The problems studied in this paper have some common properties with the above approaches since we will approach a two-dimensional problem by a family of continuous 1-dimensional problems but as each continuous 1-dimensional problem can be approximated by a discrete one, we also examine the limit of these discrete problems. The approximation of the low frequency spectrum of such problems was performed in [9, 8] (see also [17] for the plate problem), but to our best knowledge the approximation of the boundary value problem itself was not yet performed.

Hence our goal is to fill this gap and to show that indeed the solutions of the continuous and discrete 1-dimensional problems convergence to the solution of the two-dimensional problem.

The schedule of the paper is as follows: We recall in Section 2 the Dirichlet problem in the unit square as well as its continuous counterparts on networks that approach the square as

∗Universit´e de Valenciennes et du Hainaut Cambr´esis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, F-59313 - Valenciennes Cedex 9 France, email:

maryse.jospin,Serge.Nicaise,Juliette.Venel@univ-valenciennes.fr

the size goes to zero. An error estimate between the solutions of these continuous problems is proved in section 3 by using the second Strang lemma. Similarly section 4 is devoted to the error analysis between the exact solution in the unit square with the finite element approximations on the networks. In section 5 we extend some of our previous results to the Dirichlet problem set on an arbitrary domain of the plane. Finally in section 6 some numerical tests are presented that confirm our theoretical results.

Let us finish this introduction with some notations used in the remainder of the paper: On D, the L

²

(D)-norm will be denoted by k · k

_D

. The usual norm and seminorm of H

^s

(D) (s > 0) are denoted by k · k

_s,D

and | · |

_s,D

, respectively. Finally, the notation a . b means the existence of a positive constants C, which is independent of the size h of the edges of the network (see below) and of the considered quantities a and b such that a ≤ Cb.

2 The continuous problems

2.1 The continuous two-dimensional problem

Let S denote the unit square ]0; 1[×]0; 1[ and ∂S its boundary. On this domain, we consider the Dirichlet problem

−∆u = f in S

u = 0 on ∂S (1)

with f ∈ C(S).

According to Lax-Milgram lemma, there exists a unique weak solution u ∈ H

₀¹

(S ) of this problem, namely u ∈ H

₀¹

(S) is the unique solution of

Z

S

∇u · ∇v dx = Z

S

f v dx, ∀v ∈ H

₀¹

(S).

According to Theorem 5.1.3.5 of [7], this solution belongs to W

^2,p

(S), for all p > 2, and if f belongs to W

^1,p

(S), with p > 2, is such that f is zero at each corner of S , then this solution belongs to W

^3,p

(S ), hence in particular to H

³

(S).

2.2 The associated problem on networks

Now we intend to consider a similar problem set on a family of networks included in S. Let us introduce some notation to define these networks: For any n ∈ N , n ≥ 2, let h = 1/n and introduce the network R

_h

defined by

R

_h

= {]ih; (i + 1)h[×{jh}; ∀i ∈ {0, . . . , n − 1}, ∀j ∈ {1, . . . , n − 1}}

∪ {{ih}×]jh; (j + 1)h[; ∀i ∈ {1, . . . , n − 1}, ∀j ∈ {0, . . . , n − 1}}.

The edges of R

_h

are the intervals ]ih; (i + 1)h[×{jh} or {ih}×]jh; (j + 1)h[ but will be quite simply denoted by e

_i

, in other words,

R

_h

= {e

_i

; i = 1, . . . , N

_h

}, with N

_h

= 2n(n − 1).

We directly check that the size (or length) of each edge of the network R

_h

is h. We further write N

_h

for the set of nodes of R

_h

. Moreover we need to distinguish between nodes included into S or into ∂S, so we set

N

_h^int

= {(ih; jh); ∀i, j ∈ {1, . . . , n − 1}},

N

_h^ext

= {(0; ih); (1; ih); (ih; 0); (ih; 1); ∀i ∈ {0, . . . , n}},

N

_h

= N

_h^int

∪ N

_h^ext

.

It remains a last notation to indicate the set of edges adjacent to a given node:

∀v ∈ N

_h

, I

_v

= {i ∈ {1, . . . , N

_h

} such that v ∈ e

_i

}.

Our aim is to approximate the solution u of the continuous problem (1) by the solution u

_h

= (u

_i

)

_i=1,...,N

∈

N

Y

i=1

H

²

(e

_i

) of the following problem:



 

 

 

 

 

 

 

 

−u

⁰⁰_i

= f e

_i

on e

_i

∀i = 1 · · · N, u

_i

(v) = 0 ∀v ∈ N

_h^ext

, ∀i ∈ I

_v

, u

_i

(v) = u

_j

(v ) ∀v ∈ N

_h^int

, ∀i, j ∈ I

_v

, X

i∈Iv

∂u

_i

∂ν

_i

(v ) = 0 ∀v ∈ N

_h^int

,

(2)

where

f e

_i

= 1

2 γ

_i

f. (3)

In these equations,

_∂ν^∂

i

and γ

_i

represent respectively the outer normal derivative operator and the trace operator on the edge e

_i

. The last equation of problem (2) is nothing else but Kirchoff’s law. The system (2) is a Dirichlet problem on the network R

h

that was largely studied in the literature, see [13, 14, 22, 23, 12, 11, 15, 16, 19] and the references there.

2.3 Variational formulation on the networks

The variational space associated with problem (2) is V

_h

= {u

_h

= (u

_i

)

_i=1,...,N

∈

N

Y

i=1

H

¹

(e

_i

) s.t.

u

i

(v ) = u

j

(v) ∀v ∈ N

_h^int

, ∀i, j ∈ I

v

, u

_i

(v ) = 0 ∀v ∈ N

_h^ext

, ∀i ∈ I

_v

},

(4)

equipped with the norm:

||u||

_h

= |u|

1,R_h

=

"

_N

X

i=1

Z

ei

(u

⁰_i

(x))

²

dx

#

1/2

. (5)

Due to the Dirichlet boundary conditions, the H

¹

-norm and its semi-norm are equivalent on V

_h

.

Lemma 1. For every w ∈ V

h

, we have

||w||

R_h

≤ |w|

1,R_h

, (6)

as well as

kwk

_∞,Rh

:= sup

(x,y)∈R_h

|w(x, y)| ≤ |w|

_1,Rh

. (7)

Proof. Let us denote L

i

= {(x, ih), 0 < x < 1} and C

j

= {(jh, y), 0 < y < 1}. Then R

_h

=

n−1

[

i=1

L

_i

!

∪

n−1

[

j=1

C

_j

!

. (8)

As w(0, ih) = 0, we have for all x ∈]0; 1[

|w(x, ih)| =

Z

x 0

∂w

∂x (t, ih)dt

≤

∂w

∂x

_L

i

, (9)

according to the Cauchy-Schwarz inequality. Then

||w||

²_Li

= Z

1

0

|w(x, ih)|

²

dx ≤

∂w

∂x

2

Li

≤ |w|

²_1,Li

. (10) In the same way, we can check that kwk

²_C

j

≤ |w|

²_1,C

j

and by summing up these two inequalities we obtain the expected estimate (6).

The estimate (7) is a direct consequence of (9) and its counterpart in C

j

. Now we define a bilinear form a

h

on V

h

by

a

_h

: V

_h

× V

_h

→ R : (u

_h

, w

_h

) → a

_h

(u

_h

, w

_h

) =

N

X

i=1

Z

ei

u

⁰_i

(x)w

_i⁰

(x)dx, (11) that is clearly continuous and coercive on V

_h

according to Lemma 1.

Proposition 2. The variational formulation of problem (2) is to find u

_h

∈ V

_h

solution of

∀w

h

∈ V

h

, a

h

(u

h

, w

h

) = F (w

h

), (12) with

F (w

_h

) =

N

X

i=1

Z

ei

f e

_i

(x)w

_i

(x)dx. (13)

Proof. The proof is quite standard (cf. Lemma 2.2.12 in [14] for instance), we give it for the sake of completeness. Let us assume that there exists a solution u

_h

= (u

_i

)

_i=1,...,N

∈

N

Y

i=1

H

²

(e

_i

) of problem (2). Obviously u

_h

belongs to V

_h

. Moreover u

_h

is solution of (12). Indeed, let w

_h

= (w

_i

)

_i=1,...,N

∈ V

_h

, then we have for all i ∈ {1, ..., N },

− Z

ei

u

⁰⁰_i

(x)w

_i

(x)dx = Z

ei

f e

_i

(x)w

_i

(x)dx.

Integrating by parts, we obtain Z

ei

u

⁰_i

(x)w

_i⁰

(x)dx − [u

⁰_i

(v)w

_i

(v)]

^v=v_v=vⁱ²

i1

= Z

ei

f e

_i

(x)w

_i

(x)dx, (14) where v

_i1

and v

_i2

∈ N

_h

are such that i ∈ I

_vi1

∩ I

_vi2

.

We claim that

N

X

i=1

[u

⁰_i

(v)w

_i

(v)]

^v=v_v=vⁱ²

i1

= 0. (15)

In fact, we have

[u

⁰_i

(v)w

_i

(v)]

^v=v_v=vⁱ²

i1

= ∂u

_i

∂ν

_i

(v

_i1

)w

_i

(v

_i1

) + ∂u

_i

∂ν

_i

(v

_i2

)w

_i

(v

_i2

). (16) Consequently,

N

X

i=1

[u

⁰_i

(v)w

_i

(v)]

^v=v_v=vⁱ²

i1

= X

v∈N_h^ext

X

i∈Iv

∂u

_i

∂ν

_i

(v)w

_i

(v) + X

v∈N_h^int

X

i∈Iv

∂u

_i

∂ν

_i

(v)w

_i

(v).

Let v ∈ N

_h

, if v ∈ N

_h^ext

, then w

_i

(v) = 0, for all i ∈ I

_v

. If v ∈ N

_h^int

, then X

i∈Iv

∂u

i

∂ν

_i

(v)w

_i

(v ) = w

_j

(v) X

i∈Iv

∂u

i

∂ν

_i

(v) (17)

for any j ∈ I

_v

, since w

_h

is continuous at the nodes. Then, using Kirchoff’s law, the right-hand side of the identity (17) is equal to zero.

Hence (15) is established and we conclude with (14) and (15).

3 An approximation result between the continuous prob- lems

In this section, we analyze the error between the solution u of problem (1) and the solutions u

_h

of (12). For that purpose, we make use of the second Strang lemma (see below). Hence we first estimate the consistency error:

Theorem 3. Let u denote the solution of (1), and u

_h

the solution of (12). If u ∈ H

³

(S), then sup

w∈V_h

|a

_h

(u, w) − F (w)|

||w||

_h

. √

h||u||

3,S

. (18)

Proof. Since u ∈ H

³

(S), for all i = 1, ..., N , u

_i

= γ

_i

u has a meaning and since u is also continuous, its restriction to R

_h

, still denoted by u, belongs to V

_h

. Fix w = (w

_i

)

_i=1,...,N

∈ V

_h

. It can be shown, as in the proof of Proposition 2, that

a

h

(u, w) = −

N

X

i=1

Z

ei

u

⁰⁰_i

(x)w

i

(x)dx.

Then, thanks to (13),

a

_h

(u, w) − F (w) = −

N

X

i=1

Z

ei

(u

⁰⁰_i

(x) + f e

_i

(x))w

_i

(x)dx. (19)

For every v ∈ N

_h

, if (ξ, ϕ) are the coordinates of v, we define the rectangle C

_v^h

=

]ξ − h

2 , ξ + h

2 [×]ϕ − h

2 , ϕ + h 2 [

∩ S and its intersection with R

h

R

^h_v

= C

_v^h

∩ R

_h

. (20)

If v ∈ N

_h^int

, R

^h_v

is a cross, while if v ∈ N

_h^ext

, R

^h_v

is a half edge. The identity (19) can be rewritten as

a

_h

(u, w) − F (w) = − X

v∈N_h^int

Z

R^h_v

(u

⁰⁰

+ ˜ f)(x)w(x)dx − X

v∈N_h^ext

Z

R^h_v

(u

⁰⁰

+ ˜ f )(x)w(x)dx, (21)

with the abuse of notation u

⁰⁰

, that means ∂

²

u

∂x

²

or ∂

²

u

∂y

²

according to the kind of the edge (horizontal or vertical).

Step 1 : Case of the interior nodes

Fix v ∈ N

_h^int

. We define the reference square C b =] −

¹₂

; +

¹₂

[×] −

¹₂

; +

¹₂

[ and the reference cross R b = ({0}×] −

¹₂

; +

¹₂

[) ∪ (] −

¹₂

; +

¹₂

[×{0}). We consider the change of variables

φ : C b → C

_v^h

: ˆ x → x = φ(ˆ x) = v + hˆ x.

Note that φ( R) = b R

^h_v

and Z

R_v^h

(u

⁰⁰

+ ˜ f)(x)w(x)dx = Z

Rb

(u

⁰⁰

(φ(ˆ x)) + ˜ f (φ(ˆ x)))w(φ(ˆ x)) hdˆ x.

Let us set ˆ u = u ◦ φ, ˆ w = w ◦ φ, then ˆ u

⁰

= h(u

⁰

◦ φ) always with the same abuse of notation. In the same way,

u

⁰⁰

◦ φ = 1

h

²

u ˆ

⁰⁰

and (∆u) ◦ φ = 1

h

²

∆ˆ u. (22)

Owing to the definition (3) of ˜ f, ˜ f ◦ φ = −

¹_2h¹2

∆ˆ u and finally Z

R^h_v

(u

⁰⁰

+ ˜ f )(x)w(x)dx = 1 h

Z

Rb

(ˆ u

⁰⁰

− 1

2 ∆ˆ u)(ˆ x) ˆ w(ˆ x)d x ˆ = h

⁻¹

(I

₁

+ I

₂

), (23) where

I

₁

= Z

Rb

(ˆ u

⁰⁰

− 1

2 ∆ˆ u)(ˆ x)( ˆ w(ˆ x) − M w)dˆ ˆ x (24) and

I

₂

= Z

Rb

(ˆ u

⁰⁰

− 1

2 ∆ˆ u)(ˆ x)(M w)dˆ ˆ x, (25)

with the constant

M w ˆ = Z

Rb

ˆ

w(ˆ x)dˆ x. (26)

Let us begin with the estimate of I

1

. With (24) and the Cauchy-Schwarz inequality,

|I

₁

| ≤ ||ˆ u

⁰⁰

− 1

2 ∆ˆ u||

_Rb

|| w ˆ − M w|| ˆ

_Rb

. (27) Moreover we have

||ˆ u

⁰⁰

− 1 2 ∆ˆ u||

Rb

. X

|α|=2

||D

^α

u|| ˆ

Rb

. X

|α|=2

||D

^α

u|| ˆ

_1,

Cb

. (28)

by using a trace theorem [7, Thm 1.5.1.2]. We recall that due to the Poincar´ e-Friedrichs inequality,

|| w ˆ − M w|| ˆ

_Rb

. | w| ˆ

_1,bR

. (29) Thanks to (27), (28) and (29), we have shown

|I

₁

| . | w| ˆ

_1,

Rb

X

|α|=2

||D

^α

u|| ˆ

_1,

Cb

. (30)

Now in order to estimate I

₂

, we need the following lemma that can be proved by easy compu-

tations.

Lemma 4.

∀ p ˆ ∈ P

²

( R), b Z

Rb

(ˆ p

⁰⁰

− 1

2 ∆ˆ p)(ˆ x)dˆ x = 0, (31)

where P

2

( R) b represents the set of polynomials of degree at most 2 on R. b Owing to (25) and (31), for all ˆ p ∈ P

2

( R), b

I

₂

= Z

Rb

[(ˆ u − p) ˆ

⁰⁰

− 1

2 ∆(ˆ u − p)](ˆ ˆ x)(M w)d ˆ x. ˆ According to the Cauchy-Schwarz inequality, and since M w ˆ is a constant,

|I

₂

| . |M w|k(ˆ ˆ u − p) ˆ

⁰⁰

− 1

2 ∆(ˆ u − p)k ˆ

_Rb

. |M w| ˆ X

|α|=2

||D

^α

(ˆ u − p)|| ˆ

_Rb

. |M w| ˆ X

|α|=2

||D

^α

(ˆ u − p)|| ˆ

_1,

Cb

. |M w| ||ˆ ˆ u − p|| ˆ

_3,

Cb

by using the same trace theorem as previously. Let ˆ p be the orthogonal projection of ˆ u on P

2

( R) for the b H

³

( C)-norm, then b

||ˆ u − p|| ˆ

_3,Cb

. |ˆ u|

_3,Cb

. (32) Moreover, due to (26), |M w| ˆ . || w|| ˆ

Rb

, so the three last inequalities imply that

|I

₂

| . || w|| ˆ

_Rb

|ˆ u|

_3,Cb

. (33) Now we recall the next lemma that specifies the change of H

^m

-semi-norms from a domain to a reference domain [2].

Lemma 5. Consider m ∈ N and let us denote f ˆ = f ◦ φ. Then

∀f ∈ H

^m

(R

^h_v

), |f|

_m,R^h_v

= 1

h

^m−1/2

| f| ˆ

_m,Rb

and

∀f ∈ H

^m

(C

_v^h

), |f|

_m,C^h

v

= 1

h

^m−1

| f ˆ |

_m,Cb

. By (30),

|I

₁

| . | w| ˆ

_1,Rb



 X

|α|=2

||D

^α

u|| ˆ

_Cb

+ X

|α|=2

|D

^α

u| ˆ

_1,Cb



 . (34) It follows from Lemma 5 and (22) that

|I

₁

| . h

²

√

h|w|

_1,R^h_v



 X

|α|=2

h

⁻¹

||D

^α

u||

_Cv^h

+ X

|α|=2

|D

^α

u|

_1,Cv^h



 ,

and finally,

|I

₁

| . h

^3/2

|w|

_1,R^h

v

||u||

_3,C^h

v

. (35)

According to Lemma 5, (33) leads to

|I

₂

| . h

^3/2

||w||

_R^h

v

|u|

_3,C^h

v

. (36)

Gathering the results (23), (35) and (36), we have proved that

Z

R^h_v

(u

⁰⁰

+ ˜ f)(x)w(x)dx . √

h |w|

_1,Rh

v

||u||

_3,Ch

v

+ ||w||

_Rh v

|u|

_3,Ch

v

. (37) Step 2 : Case of the exterior nodes

Fix v ∈ N

_h^ext

and let us denote R b

_1/2

= φ

⁻¹

(R

^h_v

) and C b

_1/2

= φ

⁻¹

(C

_v^h

).

We show as (23) that Z

R_v^h

(u

⁰⁰

+ ˜ f)(x)w(x)dx = 1 h

Z

Rb1/2

(ˆ u

⁰⁰

− 1

2 ∆ˆ u)(ˆ x) ˆ w(ˆ x)dˆ x. (38) Using the Cauchy-Schwarz inequality, this implies

Z

R^h_v

(u

⁰⁰

+ ˜ f )(x)w(x)dx

≤ 1

h ||ˆ u

⁰⁰

− 1 2 ∆ˆ u||

_Rb

1/2

|| w|| ˆ

_Rb

1/2

. (39)

Arguing as for (28), since u ∈ H

³

(S), we get

||ˆ u

⁰⁰

− 1 2 ∆ˆ u||

Rb_1/2

. X

|α|=2

||D

^α

u|| ˆ

_1,

Cb_1/2

. (40)

On the other hand, w ∈ V

_h

implies that ˆ w(0) = 0, so it can be proved as in Lemma 1 that

|| w|| ˆ

_Rb

1/2

. | w| ˆ

_1,Rb

1/2

. (41)

Thanks to (39), (40) and (41), we have

Z

R^h_v

(u

⁰⁰

+ ˜ f )(x)w(x)dx . 1

h | w| ˆ

_1,

Rb_1/2



 X

|α|=2

||D

^α

u|| ˆ

Cb_1/2

+ X

|α|=2

|D

^α

u| ˆ

_1,

Cb_1/2



 . (42) Using Lemma 5 and the identity (22), it comes

Z

R^h_v

(u

⁰⁰

+ ˜ f)(x)w(x)dx

. 1 h

√ h|w|

_1,R^h

v

h

²



 X

|α|=2

h

⁻¹

||D

^α

u||

_C^h

v

+ X

|α|=2

|D

^α

u|

_1,C^h

v



 .

. √

h|w|

_1,R^h

v

||u||

_3,C^h

v

. (43)

Step 3 : Conclusion The identity (21) leads to

|a

_h

(u, w) − F (w)| ≤ X

v∈N_h^int

Z

R^h_v

(u

⁰⁰

+ ˜ f )(x)w(x)dx

+ X

v∈N_h^ext

Z

R^h_v

(u

⁰⁰

+ ˜ f )(x)w(x)dx

. (44) Summing (37) for all v ∈ N

_h^int

and (43) for all v ∈ N

_h^ext

, we deduce from the previous inequality

|a

_h

(u, w) − F (w)| . √

h X

v∈N_h^int

|w|

_1,R^h

v

||u||

_3,C^h

v

+ ||w||

_R^h

v

|u|

_3,C^h

v

+ √

h X

v∈N_h^ext

|w|

_1,R^h

v

||u||

_3,C^h

v

. √

h X

v∈N_h^int

||w||

_Rh v

|u|

_3,Ch

v

+ √ h X

v∈N_h

|w|

_1,Rh

v

||u||

_3,Ch v

.

By the discrete Cauchy-Schwarz inequality, we obtain

|a

_h

(u, w) − F (w)| . √

h (|w|

1,R_h

||u||

3,S

+ ||w||

R_h

|u|

3,S

) . (45)

We conclude the proof thanks to Lemma 1 and inequality (45).

Now we recall the following result which is a consequence of the second Strang Lemma and can be found for example in [2, Thm 4.2.2].

Lemma 6. Let u denote the solution of (1) supposed to belong to V

_h

, and let u

_h

be the solution of (12). Then

||u − u

_h

||

_h

. sup

wh∈Vh

|a

_h

(u, w

_h

) − F (w

_h

)|

||w

_h

||

_h

. (46)

Remark 7. Note that the upper bound in the second Strang Lemma contains in general another term, namely inf

_vh∈V_h

||u − v

_h

||

_h

, called the “interpolation error”. Here only the ”consistency error” term appears as we have assumed that u ∈ V

h

, the interpolation error being obviously equal to zero.

Corollary 8. Let u denote the solution of (1), and let u

_h

be the solution of (12). If u ∈ H

³

(S ), then

||u − u

h

||

1,R_h

.

√

h||u||

3,S

, (47)

and

||u − u

_h

||

∞,R_h

. √

h||u||

3,S

. (48)

Proof. We deduce from Theorem 3 and Lemma 6 that

|u − u

_h

|

_1,Rh

. √

h||u||

_3,S

. (49)

Now inequalities (47) and (48) are a direct consequence of Lemma 1 since u − u

_h

∈ V

_h

.

4 The finite element method on the networks

In the previous section, we have checked that u

_h

is a good approximation of u. However, problem (2) is still set in an infinite dimensional space and except for some specific right-hand sides ˜ f , its solution u

_h

cannot be computed analytically. Hence in practice problem (2) has to be discretized. Here we choose the finite element method and propose to deal with two different cases according to the regularity H

³

(S) or C

³

( ¯ S ) of the solution u.

4.1 A less regular solution

Here we assume that the solution of the continuous problem (1) u belongs to H

³

(S) and f is a continuous function in S. Let P

¹

(e

_i

) denote the set of polynomials of degree at most 1 on e

_i

, for all i ∈ {1, . . . , N }. We define the discrete variational space

W

_h

= {w

_h

= (w

_i

)

_i=1,...,N

∈ V

_h

s.t. w

_i

∈ P

1

(e

_i

), ∀i = 1, . . . , N }. (50) Let U

_h

∈ W

_h

be the solution of the finite element problem

a

_h

(U

_h

, w

_h

) = F (w

_h

), ∀w

_h

∈ W

_h

. (51) In order to compare u and U

_h

in S , we will use an interpolant I

_h

u of u and a lifting R

_h

U

_h

of U

_h

defined as follows: Let us denote K

_i,j^h

=]ih, (i + 1)h[×]jh, (j + 1)h[, for each i, j ∈ {0, . . . , n− 1}.

Observe that

S =

n−1

[

i=0 n−1

[

j=0

K

_i,j^h

!

∪ R

h

, (52)

and therefore the set of ¯ K

_i,j^h

is a triangulation of S . Hence let I

_h

u denote the Lagrange inter-

polation of u related to this triangulation, namely I

_h

u is the function such that its restriction

to K

_i,j^h

belongs to Q

1

(K

_i,j^h

) (where Q

1

is the space of polynomials in (x, y) of degree at most 1 in each variable x and y) and that coincides with u at each node v ∈ N

_h

. As a consequence I

_h

u is continuous on S . Finally R

_h

U

_h

= I

_h

U

_h

in the sense that its restriction to K

_i,j^h

fulfils R

h

U

h

∈ Q

1

(K

_i,j^h

) and

R

_h

U

_h

(v ) = U

_h

(v), ∀v ∈ K

_i,j^h

∩ N

_h

. (53) Thus R

_h

U

_h

coincides with U

_h

on R

_h

and is continuous on S .

Now we aim at approximating u by U

_h

. The estimate of the error is made with the help of the following three lemmas.

Lemma 9.

|I

_h

u − R

_h

U

_h

|

1,S

. √

h|I

_h

u − U

_h

|

1,R_h

. (54) Proof. According to Lemma 5, with ˜ Φ : ˆ C → K

_i,j^h

: ˆ x 7→ (ih, jh) + h(ˆ x +

¹₂

)

|I

_h

u − R

_h

U

_h

|

_1,K^h

i,j

. | I c

_h

u − R [

_h

U

_h

|

_1,

Cb

. (55)

Let us denote Q

⁰₁

( C) = b {q ∈ Q

₁

( C), b R

∂Cb

q = 0}.

Take q ∈ Q

₁

( C), then b

|q|

_1,Cb

= |πq|

_1,Cb

≤ kπqk

_1,Cb

, where πq = q − R

∂Cb

q ∈ Q

⁰₁

( C). Note that b Q

⁰₁

( C) is a finite dimensional space and b | · |

_1,∂

Cb

is a norm on this space. So

kπqk

_1,Cb

. |πq|

_1,∂Cb

= |q|

_1,∂Cb

. We have thus proved that

∀q ∈ Q

₁

( C), b |q|

_1,Cb

. |q|

_1,∂Cb

. (56) But I c

h

u − R [

h

U

h

∈ Q

1

( C). Thanks to (55) and (56), we have b

|I

_h

u − R

_h

U

_h

|

_1,K^h

i,j

. | I c

_h

u − R [

_h

U

_h

|

_1,∂

Cb

, and owing to Lemma 5 again,

|I

h

u − R

h

U

h

|

_1,K^h

i,j

.

√

h|I

h

u − U

h

|

_1,∂K^h

i,j

. (57)

Collecting the pieces, we obtain

|I

_h

u − R

_h

U

_h

|

²_1,S

= X

i,j

|I

_h

u − R

_h

U

_h

|

²_1,Kh i,j

. h X

i,j

|I

_h

u − U

_h

|

²_1,∂Kh i,j

. h|I

_h

u − U

_h

|

²_1,Rh

,

the last inequality following from the fact that each edge of R

_h

is in the boundary of two domains K

_i,j^h

.

Lemma 10. If u ∈ H

²

(S), then

|I

_h

u − u|

1,R_h

. √

h|u|

2,S

. (58)

Proof. Using Lemma 5,

|I

_h

u − u|

_1,∂K^h

i,j

= h

^−1/2

| I c

_h

u − b u|

_1,∂Cb

≤ h

^−1/2

|| I c

_h

u − u|| b

_1,∂Cb

. Thanks to a trace theorem (see Theorem 1.5.2.1 in [7] for instance), this leads to

|I

_h

u − u|

_1,∂K^h

i,j

. h

^−1/2

|| I c

_h

u − u|| b

_2,Cb

. (59)

By the classical interpolation error estimate (see for instance Theorem 3.1.6 in [2]), we have

|| I c

_h

u − u|| b

_2,

Cb

. |ˆ u|

_2,

Cb

. Owing to Lemma 5 again,

|| I c

_h

u − u|| b

_2,Cb

. h|u|

_2,K^h

i,j

. (60)

Then (59) and (60) imply

|I

_h

u − u|

_1,∂K^h

i,j

. √

h|u|

_2,K^h

i,j

.

We conclude the proof by squaring this inequality and summing up for i, j ∈ {0, . . . , n − 1}.

Lemma 11. Let us assume that f ∈ C(S). Then

||f ||

R_h

≤ √

2 h

^−1/2

||f||

∞,S

. (61)

Proof. Let us use the notation of the proof of Lemma 1. Then

||f ||

²_L

i

= Z

1

0

|f (x, ih)|

²

dx ≤ ||f||

²_∞,S

(62) Obviously we have the same estimate for ||f ||

²_C

j

. This leads to

||f||

²_R

h

=

n−1

X

i=1

||f||

²_Li

+

n−1

X

j=1

||f||

²_Cj

≤ 2n||f ||

²_∞,S

. Since h = 1/n, we obtain the expected result.

Proposition 12. Let u

_h

∈ V

_h

denote the solution of (12) and U

_h

∈ W

_h

the solution of (51).

Let us assume that the datum f belongs to C (S). Then

||u

_h

− U

_h

||

1,R_h

. √

h||f||

∞,S

. (63)

Proof. It can be proven (see for example Theorem 3.1.6 in [2]) that

||u

_h

− U

_h

||

_1,Rh

. h|u

_h

|

_2,Rh

. (64) But u

_h

is a solution of (2), so

|u

_h

|

2,R_h

= | f| ˜

R_h

= 1

2 ||f ||

R_h

. (65)

Due to Lemma 11,

|u

_h

|

_2,Rh

. h

^−1/2

||f ||

_∞,S

. (66)

The aim then follows from (64) and (66).

Theorem 13. Let U

h

∈ W

h

denote the solution of (51), and R

h

U

h

be defined by (53). Let us assume that the solution u of the continuous problem (1) belongs to H

³

(S), and the datum f belongs to C (S). Then

||u − R

_h

U

_h

||

1,S

. h (||u||

3,S

+ ||f||

∞,S

) . (67) Proof. As the trace of I

_h

u − R

_h

U

_h

is equal to 0 on ∂S, we have

||I

_h

u − R

_h

U

_h

||

1,S

. |I

_h

u − R

_h

U

_h

|

1,S

. (68) Lemma 9 leads to

||I

_h

u − R

_h

U

_h

||

1,S

. √

h|I

_h

u − U

_h

|

1,R_h

. √

h (|I

_h

u − u|

1,R_h

+ |u − u

_h

|

1,R_h

+ |u

_h

− U

_h

|

1,R_h

) . (69) We deduce from Lemma 10, Corollary 8 and Proposition 12 that

||I

_h

u − R

_h

U

_h

||

1,S

. h (||u||

3,S

+ ||f ||

∞,S

) . (70) On the other hand, thanks to Theorem 3.1.6 of [2],

||u − I

_h

u||

1,S

. h|u|

2,S

. (71) We conclude with (70) and (71).

4.2 A more regular solution

For more regular solutions, we will exploit the analogy with a finite difference scheme to get a pointwise convergence result.

For every v ∈ N

_h^int

, we define λ

_v

∈ W

_h

such that λ

_v

(v) = 1 and λ

_v

(v

⁰

) = 0, for all v

⁰

∈ N

_h

such that v

⁰

6= v. Remark that the support of λ

v

is included in {¯ e

i

; i ∈ I

v

} and that the set {λ

_v

, v ∈ N

_h^int

} forms a basis of the space W

_h

. The stiffness matrix M

_h

of problem (51) is easily computed. More precisely, we enumerate the interior nodes v ∈ N

_h^int

line by line, namely let us denote v

1

= (h, h), v

2

= (2h, h), . . . , v

n−1

= ((n − 1)h, h), v

n

= (h, 2h), v

n+1

= (2h, 2h), . . . , v

2n−2

= ((n − 1)h, 2h), . . . v

_(n−1)²

= ((n − 1)h, (n − 1)h). Let M

_h

denote the stiffness matrix such that

(M

h

)

i,j

= a

h

(λ

vi

, λ

vj

), ∀i, j ∈ {1, . . . , (n − 1)

²

}.

Then M

_h

is a symmetric matrix that can be written M

_h

= 1

h

A ˜

_h

(72)

where

A ˜

_h

=

















A

1,1

A

1,2

0 ... 0 A

_2,1

A

_2,2

A

_2,3

. .. 0 0 A

_3,2

A

_3,3

. .. 0 .. . . .. . .. . .. .. . 0 . . . 0 A

n−1,n−2

A

n−1,n−1

















. (73)

Each block A

_k,l

is a symmetric matrix of dimension (n − 1) and satisfy for all k ∈ {1, . . . , n −1}, A

k,k−1

= A

k−1,k

= −I

n−1

(I

n−1

is the identity matrix of dimension (n − 1)), and

A

_k,k

=











4 −1 ... 0

−1 4 . .. .. . .. . . .. ... −1 0 . . . −1 4











. (74)

Set m = (n − 1)

²

(for shortness we skip the dependence of m on h), as U

h

belongs to W

h

, it can be expressed in the basis (λ

_vk

)

_k=1...m

as follows:

U

_h

=

m

X

k=1

U

_h

(v

_k

)λ

_vk

. As usual, U

h

is the solution of problem (51) if and only if

M

h

U f

h

= F f

h

(75)

where U f

_h

= (U

_h

(v

₁

), ..., U

_h

(v

_m

))

^>

and F f

_h

= (F (λ

_v1

), ..., F (λ

_vm

))

^>

.

Now we want to check that the values of U

_h

at the nodes are a good approximation of the values of u. To this end, we observe that M

_h

is closely related to the matrix obtained by using the finite difference method to approximate the continuous problem (1). Indeed if D

_h

denote the approximation of the solution u of (1) with the finite difference method, then D

_h

is solution of the linear system [10]

A

_h

D

_h

= F

_h

, (76)

where

A

_h

= 1

h M

_h

= 1 h

²

A ˜

_h

(77)

with ˜ A

_h

defined by (73) and F

_h

= (f (v

₁

), ..., f (v

_m

))

^>

.

For further purposes, we state the following two results (see Lemma 6.2 of [1] for the proof of the first result, the second one being proved in a fully similar way, see also Property 1.20 of [20]).

Proposition 14. Let A ∈ R

^n×n

satisfying the following conditions (1) ∀i 6= j, a

_ij

≤ 0, and

(2) ∀i = 1, . . . , n, P

n

j=1

a

_ij

> 0,

then A is a monotone matrix, i. e., if X = (x

i

)

ⁿ_i=1

∈ R

ⁿ

is such that AX ≥ 0 (in the sense that (Ax)

_i

≥ 0, for all i = 1, . . . , n), then X ≥ 0.

Remark 15. The result of Proposition 14 still holds if the assumption (2) is replaced by (2’) A is a regular matrix and for all i = 1, . . . , n, P

n

j=1

a

_ij

≥ 0.

Corollary 16. A ˜

_h

and A

_h

given by (77) are monotone matrices.

Proof. Since ˜ A

_h

is symmetric and positive definite, ˜ A

_h

is a regular matrix. Moreover, ˜ A

_h

fulfils condition (1) of Proposition 14 and condition (2’) of Remark 15.

Proposition 17. Consider u the solution of Problem (1) and suppose that u ∈ C

³

(S). Set U = (u(v

₁

), ..., u(v

_m

))

^>

and let D

_h

be the solution of equation (76). Then

A

_h

(U − D

_h

) = η(u), with η(u) = (η(u)(v

₁

), ..., η(u)(v

_m

))

^>

, where

η(u)(x

_i

, y

_i

) = − h 6

∂

³

u

∂x

³

(x

_i

+ θ

_i,1

h, y

_i

) − ∂

³

u

∂x

³

(x

_i

− θ

_i,2

h, y

_i

) + ∂

³

u

∂y

³

(x

_i

, y

_i

+ θ

_i,3

h) − ∂

³

u

∂y

³

(x

_i

, y

_i

− θ

_i,4

h)

with some θ

_i,j

∈]0; 1[ and (x

_i

, y

_i

) being the coordinates of v

_i

. Moreover, one has

||η(u)||

∞

= max

i=1,...,m

|η(u)(x

_i

, y

_i

)| . hM

₃

, where M

3

= ||D

³

u||

∞

= max

(x, y) ∈ S

|α| = 3

|D

^α

u(x, y)|.