L ∞ -ERROR ESTIMATE FOR A NONCOERCIVE SYSTEM OF ELLIPTIC QUASI-VARIATIONAL

L ^∞ -ERROR ESTIMATE FOR A NONCOERCIVE SYSTEM OF ELLIPTIC QUASI-VARIATIONAL

INEQUALITIES: A SIMPLE PROOF ^∗

Messaoud Boulbrachene

Received 21 May 2004

Abstract

In this paper we provide a simple proof to derive L^∞-error estimate for a noncoercive system of quasi-variational inequalities related to the management of energy production. The key idea is a discreteL^∞-stability property owned by the corresponding coercive problem.

1 Introduction

We are interested in the piecewise linearfinite element approximation of the solution of the following system of quasi-variational inequalities (QVIs) : findU = (u¹, ..., u^J)∈ (H₀¹(Ω))^J satisfying

aⁱ(uⁱ, v−uⁱ)≥(fⁱ, v−uⁱ), ∀v∈H₀¹(Ω)

uⁱ≤M uⁱ; uⁱ≥0; v≤M uⁱ (1)

in which Ω is a bounded smooth domain of ^N where N ≥1, each aⁱ(., .) is a con- tinuous elliptic bilinear form assumed to be noncoercive, (., .) is the inner product in L²(Ω) and each fⁱ is a regular function.

This system arises in the management of energy production problems, where J power generation machines are involved, see e.g. [1], [2] and the references therein. In the case studied hereM uⁱ represents a (cost function) and the prototype encountered is

M uⁱ=k+ inf

µ=iu^µ. (2)

In (2), k represents the switching cost. It is positive when the machine is “turn on”

and equal to zero when the machine is “turn off”. Note also that operatorM provides the coupling between the unknownsu¹, ..., u^J.

∗Mathematics Subject Classifications: 65N15, 65N30.

†Department of Mathematics, College of Science, Sultan Qaboos University, P.O. Box 36 Muscat 123, Sultanate of Oman

97

In [3], we established an L^∞-error estimate for the coercive problem. This result was then extended to the noncoercive case where a subsolution approach was employed (see [4]).

In this paper we propose a new proof to derive the sameL^∞ convergence order for the noncoercive problem. This proof is much simpler than the one introduced in [4] as it rests on the sole discreteL^∞ stability property with respect to the right hand side of the solution of the corresponding coercive problem.

The paper is organized as follows. In section 2 we state both the continuous and discrete problems. In section 3 we prove a discrete L^∞ stability property for the corresponding coercive problem and give the main result of the paper.

2 Statement of the Problems

2.1 Preliminaries

We are given functions aⁱ_jk(x) inC^1,α( ¯Ω),aⁱ_k(x),aⁱ₀(x) inC^0,α( ¯Ω) such that

1≤j,k≤N

aⁱ_jk(x)ξ_jξ_k ≥α|ζ|²; ζ∈ ^N; α>0, (3)

aⁱ₀(x)≥β>0, x∈Ω. (4)

We define the bilinear form aⁱ(u, v) =

Ω

⎛

⎝

1≤j,k≤N

aⁱ_jk(x)∂u

∂x_j

∂v

∂x_k +

N

k=1

aⁱ_k(x)∂u

∂x_kv+aⁱ₀(x)uv

⎞

⎠dx. (5)

We are also given right hand side fⁱ, 1≤i≤J,such that

fⁱ∈C^0,α( ¯Ω); fⁱ≥f⁰>0. (6) Finally forW = (w¹, ..., w^J)∈(L^∞(Ω))^J we introduce the norm

W _∞= max

1≤i≤J wⁱ _L∞(Ω). (7)

2.2 The Continuous Problem

To solve the noncoercive problem, we transform (1) into the following auxiliary system:

findU = (u¹, ..., u^J)∈(H₀¹(Ω))^J such that:

bⁱ(uⁱ, v−uⁱ)≥(fⁱ+λuⁱ, v−uⁱ), ∀v∈H₀¹(Ω)

uⁱ≤M uⁱ; uⁱ≥0; v≤M uⁱ (8)

where

bⁱ(u, v) =aⁱ(u, v) +λ(v, v) (9)

andλ>0 is large enough such that

bⁱ(v, v)≥γ v ²_H1(Ω), γ >0; v∈H¹(Ω). (10) THEOREM 1 (cf. [2]). Under the preceding assumptions, there exists a unique so- lutionU = (u¹, ..., u^J) to system (1). Furthermore, this solution belongs to (W^2,p(Ω))^J for 2≤p <∞.

NOTATION 1. Throughout the paperU =∂(F+λU, M U) will denote the solution of (8) where F = (f¹, ..., f^J) andM U = (M u¹, ..., M u^J).

2.3 The Discrete Problem

Let Ωbe decomposed into triangles and letτh denote the set of all those elements;h

>0 is the mesh size. We assume that the familyτh is regular and quasi-uniform.

LetVhdenote the standard piecewise linearfinite element space andBⁱ, 1≤i≤J, be the matrices with generic coefficient

B_lsⁱ =bⁱ(ϕ_l,ϕ_s), 1≤l, s≤m(h), (11) where ϕs, s= 1,2, ..., m(h) are the nodal basis functions of the spaceVh.

Also letrh denote the usual interpolation operator.

DEFINITION. A real n×n matrix A = [aij] with aij ≤ 0 for all i = j is an M-matrix ifA is nonsingular andA⁻¹≥0.

The discrete maximum principle assumption (d.m.p): We assume that the matrices Bⁱ, are M-matrices (cf. [5]). The discrete counterpart of system (1) then reads as follows: findUh= (u¹_h, ..., u^J_h)∈(Vh)^J such that

aⁱ(uⁱ_h, v−uⁱ_h)≥(fⁱ, v−uⁱ_h), ∀v∈Vh

uⁱ_h≤rhM uⁱ_h; uⁱ_h≥0; v≤rhM uⁱ_h (12) or equivalently

bⁱ(uⁱ_h, v−uⁱ_h)≥(fⁱ+λuⁱ_h, v−uⁱ_h), ∀v∈Vh

uⁱ_h≤rhM uⁱ_h; uⁱ_h≥0; v≤rhM uⁱ_h (13) THEOREM 2 (cf. [4]). Let thed.m.phold. Then, system (12) or (13) admits a unique solution.

NOTATION 2. LetUh=∂h(F +λUh, M Uh) whereM Uh= (M u¹_h, ..., M u^J_h).

3 L

-Error Analysis

The following is a monotonicity property for the coercive problem corresponding to system (12). This result will play a crucial role in proving the discrete L^∞- stability property.

3.1 A Discrete Monotonicity Property

LetF = (F¹, ..., F^J)∈(L^∞(Ω))^J and Z_h= (z_h¹, ..., z_h^J) be the solution of the coercive system of QVIs:

bⁱ(zⁱ_h, v−zⁱ)≥(Fⁱ, v−zⁱ_h), v∈V_h

zⁱ_h≤r_hM z_hⁱ; z_hⁱ ≥0; v≤r_hM zⁱ_h . (14) Let Zh = ∂h(F, M Zh) denote that solution with z_hⁱ = σh(Fⁱ, M zⁱ_h). Following [3], existence of a unique solution can be obtained by introducing two monotone sequences:

a decreasing sequence: ¯Z_hⁿ= (¯z_h^1,n, ...,z¯_h^J,n) such that

¯

z_h^i,n+1=σh(Fⁱ, Mz¯^i,n_h )

where ¯z_h^i,0 is the unique solution of b(¯z_h^i,0, v) = (Fⁱ, v) for all v ∈ H₀¹(Ω), and an increasing sequence: Zⁿ_h= (z^1,n_h , ..., z^J,n_h ) such that

z^i,n+1_h =σ_h(Fⁱ, M z^i,n_h ) with z^i,0= 0.

THEOREM 3 (cf. [3]). The sequences ( ¯Z_hⁿ) and (Zⁿ_h) converge respectively from above and below to the unique solution of system (14).

LetF,F˜in (L^∞(Ω))^JandZh=∂h(F, M Zh), Z˜h=∂h( ˜F , MZ˜h) be the correspond- ing solutions to system (14). Then, we have the following monotonicity principle.

PROPOSITION 1. Under thed.m.p, ifF ≥F ,˜ then ∂h(F, M Zh)≥∂h( ˜F , MZ˜h).

PROOF. First, let ¯Z_h⁰ = (¯z^1,0_h , ...,z¯_h^J,0) and ˜Z⁰_h = (˜z^1,0_h , ...,z˜^,J,0_h ) be such that ¯z_h^i,0 and ˜z^i,0_h are solutions to equationsb(¯z^i,0_h , v) = (Fⁱ, v) for allv∈H₀¹(Ω) and b(˜z^i,0_h , v) = ( ˜Fⁱ, v) for all v∈H₀¹(Ω), respectively. Then, the corresponding decreasing sequences

Z¯_hⁿ= (¯z^1,n_h , ...,¯z^J,n_h ) and ˜Zⁿ_h= (˜z^1,n_h , ...,˜z^J,n_h ) satisfy the following monotonicity principle

Fⁱ ≥F˜ⁱ⇒z¯_h^i,n≥z˜^i,n_h , i= 1, ..., J.

Indeed, since

¯

z_h^i,n+1=σ(Fⁱ, Mz¯_h^i,n) and

˜

z^i,n+1_h =σ( ˜Fⁱ, Mz˜^i,n_h )

then, due to the d.m.p, Fⁱ ≥ F˜ⁱ implies ¯z_h^i,0 ≥ z˜^i,0_h , i = 1,2, ..., J. So, M z_h^i,0 ≥ M˜z^i,0_h , and thus, from thed.m.pand standard comparison results in discrete coercive variational inequalities, it follows that

¯

z_h^i,1≥z˜^i,1_h .

Now assume that ¯z^i,n_h ⁻¹ ≥z˜^i,n_h ⁻¹. Then, as Fⁱ ≥F˜ⁱ, applying the same comparison argument as before, we get

¯

z^i,n_h ≥z˜^i,n_h .

Finally, by Theorem 3, taking the the limit asn→ ∞, we getZh≥Z˜h.

3.2 A Discrete L

-Stability Property

We have the following result.

THEOREM 4. Under conditions of Proposition 1, we have

∂h(F, M Zh)−∂h( ˜F , MZ˜h)

∞≤ 1

λ+β F−F˜

∞. PROOF. Let usfirst set

Φ= 1

λ+β F−F˜

∞; Φⁱ= 1

λ+β Fⁱ−F˜ⁱ

∞. Then, we have

Fⁱ≤F˜ⁱ+ Fⁱ−F˜ⁱ

∞≤F˜ⁱ+a0(x) +λ

λ+β F−F˜ⁱ

∞. So, due to (4),

Fⁱ≤F˜ⁱ+ (a0(x) +λ)Φ.

Now making use of Proposition 1, we get

z_hⁱ ≤z˜_hⁱ +Φⁱ. Similarly, interchanging the roles of F and ˜F, we obtain

˜

z_hⁱ ≤z_hⁱ +Φⁱ. Thus

zⁱ_h−z˜ⁱ_{h L∞(Ω)}≤ Φⁱ, i= 1, ..., J, which completes the proof.

3.3 L

-Error Estimate

In what follows we prove the main result of this paper. For this purpose, we introduce the following auxiliary coercive discrete system of QVIs: find ¯Zh= (¯z¹_h, ...,z¯_h^J) solution to

bⁱ(¯zⁱ_h, v−z¯ⁱ)≥(fⁱ+λuⁱ, v−¯z_hⁱ), ∀v∈Vh

¯

zⁱ_h≤rhMz¯ⁱ_h; ¯z_hⁱ ≥0; v≤rhM¯z_hⁱ (15)

where the right hand side fⁱ+λuⁱ is the same as that of the continuous system (8).

So, the solution of system (15) is nothing else than the discrete counterpart of that of (1) or (8). Consequently, thanks to [3], we have the following error estimate.

LEMMA 1 (cf. [3]). There exists a constantC independent ofhsuch that:

U−Z¯_h

∞≤Ch²|logh|³.

THEOREM 5. Under conditions of Theorem 4 and Lemma 1, we have U−U_{h ∞}≤Ch²|logh|³.

PROOF. Indeed, since ¯Zh=∂h(F+λU, MZ¯h), then U −Uh ∞ ≤ U −Z¯h ∞+ Z¯h−Uh ∞

≤ U −Z¯_h

∞+ ∂_h(F+λU, MZ¯_h)−∂_h(F+λU_h, M U_h

∞

≤ Ch²|logh|³+ λ

λ+β U−Uh ∞

where we have used Lemma 1 and Theorem 4. Thus the desired error estimate follows.

Acknowledgment. The author would like to thank Sultan Qaboos University for the financial support under grant number: IG/SCI/DOMS/03/04. He also like to thank his colleague Dr. S. Kerbal for the help in LaTex type setting.

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