paper 17-17-2009. 305KB Jun 04 2011 12:03:07 AM

ABOUT BIHARMONIC PROBLEM VIA A SPECTRAL APPROACH AND DECOMPOSITION TECHNIQUES

Lakhdar Benaissa, Noureddine Daili

Abstract. In this work, we describe a spectral method coupled with a variational decomposition technique for solving a biharmonic equations. We construct new spaces. Using one approximation by a spectral method we can bring the resolution of Dirichlet problem for ∆2 _{to a finite number of Dirichlet}

approach for -∆. Some new theoretic spectral approaches are given, numerical solutions and illustrations are established to prove our theoretic study.

2000 Mathematics Subject Classification: 65N35, 65N30; 65N22.

Keywords and phrases: Biharmonic problem, spectral approach, approxi-mation.

1. Introduction

Let Ω be a bounded domain ofRd (d= 1,2,3 in practice) of smooth bound-ary F r(Ω). We consider the following problem:

(DP)



    

    

∆2_{u=f in Ω,}

u=g1 on F r(Ω) ;

∂u

∂η =g2 onF r(Ω),

where ∆2 _{= ∆(∆) is the biharmonic operator. The reduction of boundary}

The solution of biharmonic problem (DP) is studied by some authors and by several methods, so by finite difference methods, finite element methods and duality ...

J.W. McLaurin ([5], 1974) has given a decomposition technique for the problem (DP). He has proved that a solution of this problem by finite differ-ence methods is equivalent to resolve a sequdiffer-ence of Dirichlet problems for the operator ∆.

R. Glowinski and O. Pironneau ([4], 1977) have, only, established this de-composition method using the finite element methods correspondent to same problem on a domain of _R2_{. They have proved that a solution of this problem}

by this method is equivalent, also, to resolve a sequence of Dirichlet problems for the operator ∆ and have given the error estimate:

ku₋uhkH1_(Ω)+k∆u+ϕhkL2_(Ω) =(hk−1).

Our work consists to give a spectral approach to this problem and estimate the error theoretically on new space. We illustrate our numerical approach by numerical tests on some examples.

Indeed, if we resolve the biharmonic problem by spectral method based on the direct variational formulation, then we can give as advantage of this method the following :

i) we store one convergent approximation of the solutionuin norm_k._kH1_(Ω)

in place of the norm_k._kH2_(Ω);

ii) We obtain a convergent approximation ϕN of −∆u.

However, we will think that, in spite of the simplicity of the space VN, the practice problem is to compute (uN, ϕN).

The basic idea in this case, consists to introduce a space _MN ⊂ VN of multipliers such that

VN =V_N0 _{⊕ M}N where

V_N0 =_{vN _∈VN : vN = 0 on F r(Ω)_}

2. A Continuous Problem

Let Ω be a bounded domain of _Rd _{(d = 1,2,3 in practice) with smooth} boundary F r(Ω). Define the following problem:

(DP)           

∆2_{u=f in Ω,}

u=g1 onF r(Ω),

∂u

∂η =g2 onF r(Ω),

where g1, g2 and f are given functions. This problem is equivalent to find

(u, w) such that

(EP)                   

−∆w=f in Ω, −∆u=w on Ω, u=g1 on F r(Ω),

∂u

∂η =g2 on F r(Ω). We prove that if f _∈L2_{(Ω), g}

1 ∈H

3

2(F r(Ω)) and g₂ ∈H12(F r(Ω)) then (DP)

has one and only one solution in H2(Ω).

Theorem 1. [4] For all reals 0_≤ν _≤µ we have

ku_kHµ_(Ω)≤cN2(µ−ν)kukHν_(Ω), ∀u∈V_Nd(Ω). Theorem 2. [5] Define the space W0 as follows:

W0 =

(v, ψ)_∈H₀1(Ω)_×L2(Ω), _∀µ_∈H1(Ω), S((v, ψ), µ) = 0 ,

where

S((v, ψ), µ) =

Z

Ω∇

v_∇µ dx ₋

Z

Ω

ψµ dx.

Then, a mapping :(v, ψ)_∈W0 _{→ k}ψ_kL2_(Ω) is a norm on W0 equivalent to the

norm:

(v, ψ)_∈W0 _→(_|v_|2_H1_(Ω)+kψk

2

Further

W0 =

(v, ψ)_∈H₀2(Ω)_×L2(Ω), ₋∆v =ψ . We have

Theorem 3. [6] Let λ_∈H−12(F r(Ω)). The problem 

    

    

∆2_{u= 0 in Ω,}

−∆u=λ on F r(Ω), u= 0 on F r(Ω),

(1)

has one and only one solution in H2(Ω)_∩H₀1(Ω), and the linear operator A

defined by

Aλ=₋∂u

∂η onF r(Ω)

is an isomorphism from H−12(F r(Ω))into H12(F r(Ω)).Also, the bilinear form

a(., .) defined by

a(λ, µ) =< Aλ, µ >

where < ., . >is the bilinear form of duality between H−12(F r(Ω))and H 1

2(F r(Ω)),

is continuous, symmetric and H−12(F r(Ω))-elliptic. Namely there exist an

α >0 such that

a(µ, µ)_≥α_kµ_k2

H−12(F r(Ω)), ∀µ∈H −1

2(F r(Ω)).

We will reduce the (DP) problem to a variational equation inH−12(F r(Ω)).

Indeed, let w0 and u0 be solutions of







−∆w0 =f in Ω,

w0 = 0 on F r(Ω);

(2)







−∆u0 =w0 in Ω,

u0 =g1 on F r(Ω).

(3)

Theorem 4. [4] Let u be a solution of (DP). The trace λ of ₋∆u on

F r(Ω) is an unique solution of variational equation

< Aλ, µ >=< ∂u0

∂η −g2, µ > , ∀µ∈H

−12(F r(Ω)), λ∈H− 1

2(F r(Ω)).

Remark 1. To determine ∂u0

∂η we resolve two Dirichlet problems (2), (3), and

we resolve more again two Dirichlet problems to obtain the couple (u, w). 3. Main Results

3.1. Approximation of (DP) by Spectral Method We consider the following finite dimension space:

VN =Span{L0, L1, ..., LN} where Lk(x) are Legendre polynomials. Let

V_N0 =nvN ∈VN : vN|F r(Ω) = 0 o

and let _MN ⊂VN be such that VN =VN0 ⊕ MN. We introduce the following spaces :

WN =







(vN, ψN)_∈VN _×VN, vN_|F r(Ω) =g1, and R

Ω∇vN∇µNdx=

R

ΩψNµNdx+

R

F r(Ω)g2µNdσ, ∀µN ∈VN







;

W_N0 =

(vN, ψN)_∈V_N0 _×VN,

Z

Ω∇

vN_∇µNdx ₋

Z

Ω

ψNµNdx= 0, _∀µN _∈VN

.

3.1.1. Choice of V_N0 and Appropriate Basis for Galerkin Method If we have a nonhomogenous Dirichlet problem associated to the operator (∆) and posed on the space VN, we can define one transformation for which this problem will be equivalent to one posed homogenous problem onV0

N. The Galerkin approach consists plainly to replace the test functions space by polynomials space.

i) the way from which space V_N0(Ω) approaches the space V; ii) steepness and the simpleness calculus of coefficientsaij and Fj; iii) steepness to resolve a linear systemAu =F.

To satisfy the 1st criterion i), we will consider the space V_N0(Ω) of enough large dimension.

To satisfy the 2nd and 3rd critera ii) and iii), it will require obtain one sufficiently deep matrixAsuch that the linear system Au=F does not enough at cost (in time and required space machine), and such that there are not coefficients aij to compute.

What is to be done in the choice of a basis of V_N0(Ω) such that the linear system to resolve will be possible ?

To answer this question, we need the following lemma : Lemma 1. [6] Put

ck = _√ 1

4k+6, φk(x) =ck(Lk(x)−Lk+2(x))

ajk = (φ

′

k(x), φ

′

j(x)) , bjk = (φk(x), φj(x)), k, j = 0,1, ..., N −2.

Then,

ajk =







1 if k=j, 0 if k ₆=j,

bkj =bjk =



    

    

ckcj( 2 2j+1 +

2

2j+5), if k =j,

−ckcj, if k=j+ 2,

0 else,

and

V_N0(I) = Span_{φ0(x), φ1(x), ..., φN−2(x)}, where (., .) is an L2_{-inner product.}

3.1.2. Choice of _MN

Let φ₋1, φN−1 be two functions such that φ−1, φN−1 ∈ VN and φ−1,φ0,

φ1, ..., φN−2, φN−1 are linearly independent.

Proposition 1. For d= 1, the space MN is given by MN =Span{φ−1(x), φN−1(x)}. For d= 2, the space MN is given by

MN =Span







φ₋1(x)φj(y), φN−1(x)φj(y), −1≤j ≤N −1

φi(x)φ−1(y), φi(x)φN−1(y), 0≤i≤N −2







.

For d= 3, the space MN is given by

MN =Span

              

φi(x)φj(y)φ−1(z), φi(x)φj(y)φN−1(z), −1≤i, j ≤N −1;

φi(x)φ₋1(y)φk(z), φi(x)φN−1(y)φk(z), −1≤i≤N −1,

0_≤k _≤N ₋2; φ₋1(x)φj(y)φk(z), φN−1(x)φj(y)φk(z), 0≤j, k≤N −2

               .

Remark 2. The choice of functions φ₋1, φN−1 is not unique, then the choice of space MN is not unique.

Now, we give some choices of φ₋1, φN−1:

Remark 3. i) φ₋1 =L0, φN−1 =L1 ii) φ₋1 =L0, φN−1 =LN−1−L1 iii) φ₋1 =L1−L0, φN−1 =LN−1.

Numerical tests are proofs to the good choice. We approache then (DP) by

(DP)N







Find (uN, ϕN)_∈WN such that

jN(uN, ϕN)≤jN(vN, ψN), ∀(vN, ψN)∈WN, where

jN(vN, ψN) = 1 2

Z

Ω|

ψN|2dx −

Z

Ω

f vNdx.

4. Convergence Results

Consider the following problem:

(DP)0

          

∆2_{u=g in Ω,}

u= 0 on F r(Ω), ∂u

∂η = 0 on F r(Ω), which is equivalent to the following optimization problem:

(DP)1







Find u_∈H2

0(Ω) such that

J0(u)≤J0(v), ∀v ∈H02(Ω),

where

J0(v) =

1 2

Z

Ω|

∆v_|2dx ₋

Z

Ω

gvdx.

Now, we prove some results of convergence.

Lemma 2. Let u be a solution of problem (DP)0. Then there exist a constant c >0 independent of N such that

|u₋uN_|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤c( inf

(vN, ψN)∈W0

N

(_|u₋vN_|H1_(Ω)+k∆u+ψNkL2_(Ω))

+ inf

µN∈VNk∆u+µNkH1(Ω)). Proof. Letu be a solution of problem (DP)0, then

−

Z

Ω∇

v_∇(∆u)dx=

Z

Ω

∆u∆vdx =

Z

Ω

gvdx, _∀v _{∈ D}(Ω) but _D(Ω) is dense in H₀1(Ω), then

−

Z

Ω∇

v_∇(∆u)dx=

Z

Ω

gvdx, _∀v _∈H₀1(Ω). We have

S((v, ψ), µ) =

Z

Ω∇

v_∇µdx ₋

Z

Ω

and

S((v, ψ),₋∆u) = ₋

Z

Ω∇

v_∇(∆u)dx +

Z Ω ψ∆udx= Z Ω gvdx + Z Ω ψ∆udx.

Let (vN, ψN)_∈W_N0 and µN _∈VN , then S((uN ₋vN, ϕN ₋ψN),∆u+µN) =R

Ω∇(uN −vN)∇(∆u+µN)dx

−R

Ω(ϕN −ψN)(∆u+µN)dx

=S((uN −vN, ϕN −ψN), ∆u) +S((uN −vN, ϕN −ψN), µN) =₋R

Ωg(uN −vN)dx −

R

Ω(ϕN −ψN)∆udx

+R

Ω∇(uN −vN)∇µNdx −

R

Ω(ϕN −ψN)µNdx

=₋R

Ω(ϕN −ψN)∆udx −

R

ΩguNdx +

R

ΩgvNdx

+ R

Ω∇uN∇µNdx−

R

Ω∇vN∇µNdx −

R

ΩϕNµNdx+

R

ΩψNµNdx.

Or

Z

Ω∇

uN_∇µNdx=

Z

Ω

ϕNµNdx and

Z

Ω

ϕNψNdx=

Z Ω gvNdx, then Z Ω

(ϕN ₋ψN)(∆u+ϕN)dx=₋S((uN ₋vN, ϕN ₋ψN), ∆u+µN) By continuity of S((., .), .), we have

R

Ω(ϕN −ψN)(∆u+ϕN)dx

≤c(|uN −vN|H1_(Ω)k∆u+µNkH1_(Ω)

+_kϕN ₋ψN_kL2_(Ω)k∆u+µNkH1_(Ω))

Using Theorem 2, it holds,

and

R

Ω(ϕN −ψN)(∆u+ϕN)dx

≤c(c(Ω)kϕN −ψNkL2_(Ω)k∆u+µNkH1_(Ω)

+_kϕN −ψNkL2_(Ω)k∆u+µNkH1_(Ω))

≤d(Ω)_kϕN −ψNkL2_(Ω)k∆u+µNkH1_(Ω),

where d(Ω) = max(cc(Ω), c). Thus kϕN −ψNk2L2_(Ω) =

R

Ω(ϕN −ψN)(∆u+ϕN)dx −

R

Ω(ϕN −ψN)(∆u+ψN)dx

kϕN −ψNk2L2_(Ω) ≤d(Ω)kϕN −ψNkL2_(Ω)k∆u+µNkH1_(Ω)

+_kϕN −ψNkL2_(Ω)k∆u+ψNkL2_(Ω).

It holds that

kϕN ₋ψN_kL2_(Ω) ≤d(Ω)k∆u+µNkH1_(Ω)+k∆u+ψNkL2_(Ω) (4)

and

|u₋uN|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤ |u−vN|H1_(Ω)+|vN −uN|H1_(Ω)

+_k∆u+ψNkL2_(Ω)+kϕN −ψNkL2_(Ω)

≤ |u₋vN_|H1_(Ω)+k∆u+ψNkL2_(Ω)+ (1 +c(Ω))kϕN −ψNkL2_(Ω) .

Using inequality (4), we obtain

|u₋uN|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤(|u−vN|H1_(Ω)+k∆u+ψNkL2_(Ω)

+(1 +c(Ω)))_·(d(Ω)_k∆u+µNkH1_(Ω)+k∆u+ψNkL2_(Ω)),

hence

|u₋uN_|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤ |u−vN|H1_(Ω)+ (2 +c(Ω))k∆u+ψNkL2_(Ω)

+(1 +c(Ω))d(Ω)_k∆u+µN_kH1_(Ω)

where

c∗ = max(2 +c(Ω),(1 +c(Ω))d(Ω)). Therefore, it holds

|u₋uN|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤c∗( inf

(vN,ψN)∈W0

N

(_|u₋vN|H1_(Ω)+k∆u+ψNkL2_(Ω))

+ inf

µN∈VNk∆u+µNkH1(Ω)). Lemma 3. Let u _∈H2(Ω) be a solution of optimization problem (DP)0. If u _∈ Hk+2_{(Ω), k ≥ 2, then there exist a constant c > 0 independent of N} such that

|u₋uN_|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤cN1−k(|u|Hk+2_(Ω)+|∆u|Hk_(Ω)) Proof. Let (vN, ψN)_∈W_N0 and µN _∈VN. Put

νN =µN +ψN, νN ∈VN. Then, we have

S((vN, ψN), νN) = 0, (5)

Z

Ω

∆uνNdx=₋

Z

Ω∇

u_∇νNdx +

Z

F r(Ω)

∂u ∂ηνNdσ

Or ∂u

∂η |F r(Ω)= 0, so one has

Z

Ω

∆uνNdx=₋

Z

Ω∇

u_∇νNdx. (6) By (5) and (6), we have

Z

Ω

(∆u+ψN)νNdx=

Z

Ω∇

(vN ₋u)_∇νNdx

Z

Ω

(∆u+ψN)νNdx

≤ |

u₋vN_|H1_(Ω)|νN|H1_(Ω).

By Theorem 1, we obtain

so

Z

Ω

(∆u+ψN)νNdx

≤

cN2_|u₋vN|H1_(Ω)kνNkL2_(Ω)

And

kνNk2L2_(Ω)= R

Ω(µN −∆u)νNdx+

R

Ω(∆u+ψN)νNdx

≤ k∆u₋µNkL2_(Ω)kνNkL2_(Ω)+cN2|u−vN|H1_(Ω)kνNkL2_(Ω)

⇒ kνN_kL2_(Ω) ≤ k∆u−µNkL2_(Ω)+cN2|u−vN|H1_(Ω).

Hence

k∆u+ψN_kL2_(Ω) =k∆u+νN −µNkL2_(Ω)

≤ k∆u₋µN_kL2_(Ω)+kνNkL2_(Ω)≤2k∆u−µNkL2_(Ω)+cN2|u−vN|H1_(Ω).

Then

|u₋vN|H1_(Ω)+k∆u+ψNkL2_(Ω) ≤(1 +cN2)|u−vN|H1_(Ω)+ 2k∆u−µNkL2_(Ω),

∀(vN, ψN)∈WN0 , ∀µN ∈VN. Thus

inf

(vN,ψN)∈W0

N

(_|u₋vN_|H1_(Ω)+k∆u+ψNkL2_(Ω))≤(1 +cN2) inf

vN∈V0

N

|u₋vN_|H1_(Ω)

+2 inf

µN∈VNk∆u−µNkL2(Ω). Lemma 2 implies

|u₋uN|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤c((1 +cN2) inf

vN∈V0

N

|u₋vN|H1_(Ω)

+ inf

µN∈VNk∆u−µNkH1(Ω)). Further, we have

inf vN∈V0

N

|u₋vN_|H1_(Ω) ≤cN−1−k|u|Hk+2_(Ω)

inf

µN∈VNk∆u+µNkH1(Ω) ≤cN

then, it holds

|u₋uN|H1_(Ω)+k∆u+ϕNkL2_(Ω) ≤c((1 +cN2)cN−1−k|u|Hk+2_(Ω)+cN1−k|∆u|Hk_(Ω)) ≤cN1−k_(|u|H

k+2_(Ω)+|∆u|Hk_(Ω)). Theorem 5. Le u_∈H2_{(Ω) be a solution of a problem (DP), then for all} integer k _≥2, if u_∈Hk+2(Ω) we have

ku₋uN_kH1_(Ω)+k∆u+ϕNkL2_(Ω) ≤cN1−kkukHk+2_(Ω)

If u_∈Hk+1(Ω), k _≥3,then

ku₋uNkL2_(Ω)+

1

N2 k∆u+ϕNkL2(Ω)≤cN−1−kkukHk+1_(Ω)

Proof. We will reduce the problem (P D)N to one variational problem in MN. Indeed, let aN(., .) : MN × MN → R be a bilinear form defined for λN ∈ MN by

Z

Ω∇

wN_∇vNdx= 0 , _∀vN _∈V_N0 , wN ₋λN _∈V_N0; (7)

Z

Ω∇

uN_∇vNdx=

Z

Ω

wNvNdx , _∀vN _∈V_N0 , uN _∈V_N0; (8)

aN(λN, µN) =

Z

Ω

wNµNdx−

Z

Ω∇

uN∇µNdx, ∀µN ∈ MN. (9) Lemma 4. A bilinear form aN(., .) is positive definite and symmetric on

MN × MN. Moreover

aN(λ1N, λ2N) =

Z

Ω

w1Nw2Ndx, ∀λ1N, λ2N ∈ MN,

where w1N, w2N are solutions of (7) associated to λ1N, λ2N. Proof. By definition one has

aN(λ1N, λ2N) =

Z

Ω

w1Nλ2Ndx −

Z

Ω∇

u1N∇λ2Ndx, ∀µN ∈ MN,

Put λ2N = (λ2N −w2N) +w2N, then aN(λ1N, λ2N) =

R

Ωw1Nw2N dx −

R

Ω∇u1N∇w2Ndx+

R

Ω∇u1N∇(w2N −λ2N)dx

−R

Ωw1N∇(w2N −λ2N)dx

One hasu1N ∈VN0, using (7) one has

Z

Ω∇

u1N∇(w2N −λ2N)dx=

Z

Ω

w1N∇(w2N −λ2N)dx. Then

aN(λ1N, λ2N) =

Z

Ω

w1Nw2Ndx, ∀λ1N, λ2N ∈ MN.

It is obvious that aN(., .) is symmetric and coercive.

Theorem 6. Let (uN, wN) be a solution of (DP)N and λN the compo-nent of wN in MN. Then λN is an unique solution of the linear variational

problem:

aN(λN, µN) = R

Ω∇u0N∇µNdx −

R

Ωw0NµNdx −

R

F r(Ω)g2NµNdσ,

∀µN _{∈ M}N , λN ∈ MN

(10)

where w0N is a solution of

Z

Ω∇

w0N ∇vNdx=

Z

Ω

f vNdx , ∀vN ∈VN0 , w0N ∈VN0 (11)

and u0N is a solution of





 R

Ω∇u0N∇vNdx=

R

Ωw0N vNdx , ∀vN ∈V 0

N ,

u0N =g1 on F r(Ω).

(12)

Proof. We have aN(λN, µN) =R

ΩwN¯ µNdx−

R

Ω∇uN¯ ∇µNdx

=R

Ω(wN −w0N)µNdx−

R

Ω∇(uN −u0N)∇µNdx

=R

Ω∇u0N∇µNdx −

R

Ωw0NµNdx−(

R

Ω∇uN∇µNdx−

R

But (uN, wN)∈WN, hence

Z

Ω∇

uN_∇µNdx₋

Z

Ω

wNµNdx=

Z

F r(Ω)

g2NµNdσ, ∀µN ∈ MN

Therefore aN(λN, µN) =

Z

Ω∇

u0N∇µNdx−

Z

Ω

w0NµNdx−

Z

F r(Ω)

g2NµNdσ, ∀µN ∈ MN.

Existence and uniqueness of a solution of (10) is a direct consequence from Lax-Milgram Lemma. A system is symmetric because a formaN(., .) is symmetric. Remak 4. The problem (10) is equivalent to a system from which associ-ated matrix is positive definite and symmetric.

Now we prove a problem (10) is equivalent to a system from which associ-ated matrix is positive definite.

5. Numerical Approach

5.1. Construction of Linear System Associated to the Variational Problem (10)

Put lN(µN) =

Z

Ω∇

u0N∇µNdx −

Z

Ω

w0NµNdx −

Z

F r(Ω)

g2µN dσ, ∀µN ∈ MN

Then solve a problem (10) is equivalent to resolve the following system : aN(λN, µN) =lN(µN), _∀µN _{∈ M}N, λN ∈ MN . (13) Define the set

Id=

                                  

{−1, N₋1_} if d= 1







(₋1, j),(N ₋1, j), ₋1_≤j _≤N₋1 (i, ₋1), (i, N ₋1), 0_≤i_≤N₋2







, if d= 2

          

(i, j, ₋1), (i, j, N ₋1),₋1_≤i, j _≤N ₋1,

(i, ₋1, k), (i, N₋1, k) , ₋1_≤i_≤N ₋1,0_≤k _≤N ₋2 (₋1, j, k), (N ₋1, j, k), 0_≤j, k_≤N ₋2

          

and

φid =φi₁φi₂ ·...·φid d = 1, 2,3 where id_{= (i}

1, i2, ..., id). So (13) is equivalent to



 

 

aN(λN, φid) =l_N(φ_id), id∈I_d λN = P

jd_∈Id

λ_jdφ_jd.

(14)

Namely

X

jd_∈Id

λ_jda_N(φ_jd, φ_kd) = l_N(φ_kd), kd∈I_d (15) and

p=Card(Id) = dim(_MN) =



    

    

2, if d= 1,

4N, if d= 2,

6N2_{+ 2, if d= 3.}

5.2. Computation of the Matrix AN

Denote _BN the basis of MN and wjd_N, u_jd_N solutions respectively of

R

Ω∇wjdN∇vNdx= 0, ∀vN ∈V 0

N, wjd_N ∈VN, w_jd_N −φ_jd ∈V_N0 ;

R

Ω∇ujd_N∇v_Ndx=

R

Ωwjd_Nv_Ndx, ∀v_N ∈V_N0 , u_jd_N ∈V_N0, further we have

a_id_jd =a_N(φ_jd, φ_id) =

Z

Ω

w_jd_Nφ_iddx−

Z

Ω∇

u_jd_N∇φ_iddx, id, jd∈I_d.

It results that a computation of a matrix AN requires the computation of 2p Dirichlet problems for an operator ∆.

We have

Theorem 7. For all integer N _≥1, we have

α 1

N kγ0λNk

2

L2_{(F r(Ω))} ≤aN(λN, λN)≤βkγ0λNk2_L2_{(F r(Ω))}, ∀λN ∈ MN,

Proof. LetλN ∈ MN, we have aN(λN, λN) =

Z

Ω

w2_Ndx,

where wN is a solution of (7). Let ˜wN ∈H1(Ω) be a solution of







∆ ˜wN = 0 in Ω, ˜

wN =λN on F r(Ω), then one has

p

aN(λN, λN) =kwNkL2_(Ω)

≤ kwN −w˜NkL2_(Ω)+kw˜NkL2_(Ω) =kwN −w˜NkL2_(Ω)+

√

< Aγ0λN, γ0λN >.

Put_|A_|=_kA_kLc(L2_{(F r(Ω)),L}2_{(F r(Ω)))}, it holds that

|< Aλ, µ >_{| ≤ |}A_{| k}λ_kL2_{(F r(Ω))}kµkL2_{(F r(Ω))}, ∀λ, µ∈L2(F r(Ω))

⇒p

aN(λN, λN)_{≤ k}wN ₋wN˜ _kL2_(Ω)+ p

|A_{| k}γ0λNkL2_{(F r(Ω))}.

and we have

kwN ₋wN˜ _kL2_(Ω) ≤cN− 3 2 kwN˜ k

H32(Ω),

and

kw˜Nk_H3

2(Ω) ≤ckλNkH1(F r(Ω))

kλN_kH1_{(F r(Ω))} ≤cNkλNkL2_{(F r(Ω))}.

Namely

p

aN(λN, λN)≤ckγ0λNkL2_{(F r(Ω))}.

And

c1

N kγ0vNk

2

L2_{(F r(Ω))} ≤ kvNkL22_(Ω), ∀vN ∈VN. It holds that

aN(λN, λN) =kwNk2L2_(Ω) ≥c

1

N kγ0wNk

2

L2_{(F r(Ω))} =c

1

N kγ0λNk

2

Proposition 2. (Condition Number of the Matrix AN) We have cond(A) =O(N),

where A is a square matrix associated to the problem (10).

Proof. LetA be a square inversible matrix N _×N. We have cond(A) = _kA_k.A−1

,

where _k._k is a matrix norm associated to canonical euclidean norm of _RN_{. If} A is positive definite and symmetric then

cond(A) = λmax λmin

,

where λmax is the largest eigenvalue and λmin is the smallest eigenvalue of A.

By Theorem 7, there exist two constants α and β >0 such that α 1

N kγ0λNk

2

L2_{(F r(Ω))} ≤aN(λN, λN)≤βkγ0λNk2L2_{(F r(Ω))}, ∀λN ∈ MN. Put

pmax = sup

λN∈MN−{0}

aN(λN, λN) kγ0λNk2_{L2(F r(Ω))}

pmin = inf

λN∈MN−{0}

aN(λN, λN) kγ0λNk2_{L2(F r(Ω))},

then we have

pmax ≤β and pmin ≥α

1 N,

where pmax is the largest eigenvale and pmin is the smallest eigenvalue ofAN.

Therefore, we have 1 pmin ≤

N α ⇒

pmax

pmin ≤

cN _⇒cond(A) = (N)

5.3. Algorithm (Conjugate Gradient Method Applied to Varia-tional Problem (10)

We introduce the isomorphism rN :MN →Rp defined as follows : µN ∈ MN, µN =

p

X

i=1

rN ={µ1, µ2, ..., µP}, ∀µN ∈ MN. Then

aN(λN, µN) = (ANrNλN, rNµN)N, ∀λN, µN ∈ MN and

Z

Ω∇

u0N∇µNdx−

Z

Ω

w0NµNdx−

Z

F r(Ω)

g2NµNdσ= (bN, rNµN)N, ∀µN ∈ MN,

where (., .)N is the inner euclidien scalar of Rp and _k._kN is the associated norm.

Algorithm :

Step 1: k = 0, let λ0

N ∈ MN be an arbitrary initial data. g_N0 =ANrNλ0_{N −}bN , d0_N =g_N0

Step 2 : If _kg_N0_{kN ≤}ε stop, else put ρn= (dnN, gnN)N

(ANdn N, d

n N)N

rNλnN+1 =rNλnN −ρndnN g_Nn+1 =gn

N −ρnANdnN and

βn =

(gn+1

N , g n+1

N )N

(gn N, gnN)N

dn_N+1 =g_Nn+1+βndnN+1 Step 3 : k _←k+ 1 and return to step 2. Numerical Results :

Example 1. Consider the following problem :



      

      

∆2_u=−128π4_{cos(4πx) in Ω = ]−1,+1[}2

,

u= 0 on F r(Ω), ∂u

[image:20.612.196.400.92.221.2]

Figure 1: exact solutionu(x) = (sin(2πx))2

This problem has one and only one solution : u(x, y) = (sin(2πx))2_.

We have:

If we select the 3 rd choice of the space _MN with







φ₋1 =L1−L0,

φN₋1 =LN−1,

then we have

[image:20.612.200.386.318.533.2]

[image:21.612.199.390.103.227.2] [image:21.612.196.393.289.411.2]

Figure 3: Comparison between exact and approach solutions for N=12

Figure 4: approach solution for N=14

[image:21.612.204.393.472.592.2]

[image:22.612.204.394.143.268.2]

Figure 6: approach solution for N=16

[image:22.612.200.396.417.544.2]

Example 2. Consider the following problem :



      

      

∆2_{u= 24(1−x}2₎2_{+ 24(1−y}2₎2_{+ 32(3x}2_−1)(3y2_{−1) in Ω = ]−1,+1[}2

,

u= 0 on F r(Ω), ∂u

∂η = 0 on F r(Ω).

This problem has one and only one solution: u(x, y) = (1₋x2)2_(1−y2₎2_{. We}

[image:23.612.199.391.267.416.2]

have

[image:24.612.205.373.94.254.2]

Figure 9: exact solution with contour

If we select the 1st choice of the space _MN with

φ₋1 =L0,

φN−1 =L1,

then we have

[image:24.612.205.374.322.558.2]

[image:25.612.202.373.89.252.2]

Figure 11: approach solution with lines of contour for N=4

6. Conclusion

We will say that developped methods are globally interesting when we dispose beforehand a good and well program to resolve the Dirichlet approach problem. One generalization of this problem is given in ([2]). An other more intersting problem is to study the evolution problem and the estimate of error for this last.

References

[1] L. Benaissa and N. Daili,Spectral Approximations of Stationary Dirich-let Problem for Harmonic Operator, Submited (2008), pp. 1-22.

[2] L. Benaissa and N. Daili,Spectral Approximations of Stationary Dirich-let Problem for Polyharmonic Operator and Decompositions, Preprint (2008), pp. 1-50.

[3] C. Bernardi-Y. Maday, Approximation Spectrale de Probl`emes aux lim-ites elliptiques, Springer-Verlag, Berlin, 199.

[4] R. Glowinski - O. Pironneau, Sur une m´ethode Quasi-Directe pour l’Op´erateur Biharmonique et ses Applications `a la R´esolution des Equations de Navier-Stokes, Ann. Sc. Math. Qu´ebec, Vol.1, No. 2, (1977), pp. 231-245. [5] J. W. McLaurin, A general coupled equation approach for solving, Siam. Jour. Num. Anal.11 (1974), pp. 14-33.

(1994), pp. 1489-1505. Authors:

Lakhdar Benaissa

Hight school of commerce.

1 Rampe Salah Gharbi. Algiers. Algeria. email: l benaissa@esc-alger.com

Noureddine Daili

Department of Mathematics. University of Ferhat Abbas. 19000 S´etif-Algeria