International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 703670 | 15 pages | https://doi.org/10.5402/2011/703670

A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities

Academic Editor: S. Zhang
Received30 Mar 2011
Accepted13 May 2011
Published07 Jul 2011

Abstract

The paper deals with the theta time scheme combined with a finite element spatial approximation of parabolic variational inequalities. The parabolic variational inequalities are transformed into noncoercive elliptic variational inequalities. A simple result to time energy behavior is proved, and a new iterative discrete algorithm is proposed to show the existence and uniqueness. Moreover, its convergence is established. Furthermore, a simple proof to asymptotic behavior in uniform norm is given.

1. Introduction

A great work has been done on questions of existence and uniqueness for parabolic variational and quasivariational inequalities over the last three decades. However, very much remains to be done on the numerical analysis side, especially error estimates and asymptotic behavior for the free boundary problems (cf., e.g., [1–8]).

In this paper, we propose a new iterative discrete algorithm to prove the existence and uniqueness, and we devote the asymptotic behavior using the 𝜃 time scheme combined with a finite element spatial approximation for parabolic inequalities.

Let us assume that 𝐾 is an implicit convex set defined as follows: 𝐾=𝑣(𝑡,𝑥)∈𝐿20,𝑇,𝐻10(Ω),𝑣(𝑡,𝑥)≤𝜓(𝑡,𝑥),𝑣(0,𝑥)=𝑣0inΩ,(1.1) with 𝜓∈𝐿20,𝑇,𝑊2,∞(Ω).(1.2)

We consider the following problem, find 𝑢∈𝐾 solution of 𝜕𝑢𝜕𝑡+𝐴𝑢≤𝑓inΣ,𝑢(𝑡,𝑥)=0inΓ,(1.3) where Σ is a set in ℝ×ℝ𝑁 defined as Σ=Ω×[0,𝑇] with 𝑇⋅<+∞, and Ω is convex domain in ℝ𝑁, with sufficiently smooth boundary Γ.

The symbol (⋅,⋅) stands for the inner product in 𝐿2(Ω), and 𝐴 is an operator defined over 𝐻1(Ω) by 𝐴𝑢=−𝑁𝑖𝑗=1ğœ•ğœ•ğ‘¥ğ‘–ğ‘Žğ‘–ğ‘—(𝑥)𝜕𝑢𝜕𝑥𝑗+𝑁𝑗=1𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗+ğ‘Ž0(𝑥)𝑢,(1.4) and whose coefficients: ğ‘Žğ‘–,𝑗(𝑥),𝑏𝑗(𝑥),ğ‘Ž0(𝑥)âˆˆğ¿âˆž(Ω)∩𝐶2(Ω),𝑥∈Ω,1≤𝑖,𝑗≤𝑁 are sufficiently smooth functions and satisfy the following conditions: ğ‘Žğ‘–ğ‘—(𝑥)=ğ‘Žğ‘—ğ‘–(𝑥);ğ‘Ž0(𝑥)≥𝛽>0,𝛽isaconstant,(1.5)𝑁𝑖𝑗=1ğ‘Žğ‘–ğ‘—(𝑥)𝜉𝑖𝜉𝑗||𝜉||≥𝛾2;𝜉∈ℝ𝑁,𝛾>0,𝑥∈Ω,(1.6)𝑓 is a regular functions satisfying 𝑓∈𝐿2(0,𝑇,ğ¿âˆž(Ω))∩𝐶10,𝑇,𝐻−1(Ω),𝑓≥0.(1.7)

We specify the following notations: ‖⋅‖𝐿2(Ω)=‖⋅‖2,‖⋅‖1=‖⋅‖𝐻10(Ω),â€–â‹…â€–ğ¿âˆž(Ω)=‖⋅‖∞.(1.8)

As we have said before, the aim of the present paper is to show that the asymptotic behavior can be properly approximated by a 𝜃 time scheme combined with a finite element spatial using a new iterative algorithm. We precede our analysis in two steps: in the first step, we discretize in space; that is, we approach the space 𝐻10 by a space discretization of finite dimensional ğ‘‰â„ŽâŠ‚ğ»10. In the second step, we discretize the problem with respect to time using the 𝜃-scheme. Therefore, we search a sequence of elements ğ‘¢ğ‘›â„Žâˆˆğ‘‰â„Ž which approaches 𝑢𝑛(𝑡𝑛),𝑡𝑛=𝑛Δ𝑡, with initial data 𝑢0ℎ=𝑢0ℎ. Our approach stands on a discrete stability result and error estimate for parabolic variational inequalities.

The paper is organized as follows. In Section 2, we prove the simple result to time energy behavior of the semidiscrete parabolic variational inequalities. In Section 3, we prove the ğ¿âˆž-stability analysis of the 𝜃-scheme for P.V.I, and finally, in Section 4, we first associate with the discrete P.V.I problem a fixed point mapping, and we use that in proving the existence of a unique discrete solution, and later, we establish the asymptotic behavior estimate of 𝜃-scheme by the uniform norm for the problem studied.

2. Priory Estimate of the Discrete Parabolic Variational Inequalities

We can reformulate (1.3) to the following variational inequality: 𝜕𝑢𝜕𝑡,𝑣−𝑢+ğ‘Ž(𝑢,𝑣−𝑢)≥(𝑓,𝑣−𝑢),𝑣∈𝐾,(2.1) where ğ‘Ž(⋅,⋅) is the bilinear form associated with operator 𝐴 defined in (1.4). Namely, î€œğ‘Ž(𝑢,𝑣)=Ω𝑁𝑖𝑗=1ğ‘Žğ‘–ğ‘—(𝑥)𝜕𝑢𝜕𝑥𝑖𝜕𝑣𝜕𝑥𝑗+𝑁𝑗=1𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗𝑣+ğ‘Ž0(𝑥)𝑢𝑣𝑑𝑥,(2.2)

Theorem 2.1 (see [9]). The problem (1.3) has an unique solution 𝑢∈𝐾(𝑢). Moreover, one has 𝑢∈𝐿20,𝑇;𝐻10,(Ω)𝜕𝑢𝜕𝑡∈𝐿20,𝑇;𝐻−1(Ω).(2.3)

Lemma 2.2 (Sobolev-Poincare inequality). Let Ω be a bounded overt in ℝ𝑁, with sufficiently smooth boundary Γ, then there exists a 𝐶∗ such that ‖𝑢‖2≤𝐶∗‖∇𝑢‖2,𝑣∈𝐻10(Ω)∩𝐶2Ω,∇=𝑁𝑖=1𝜕𝜕𝑥𝑖.(2.4)

2.1. The Discrete Problem

Let us assume that Ω can be decomposed into triangles and ğœâ„Ž denotes the set of all the elements ℎ>0, where ℎ is the mesh size. We assume that the family ğœâ„Ž is regular and quasi-uniform, and we consider the usual basis of affine functions 𝜑𝑖, 𝑖={1,…,𝑚(ℎ)} defined by 𝜑𝑖(𝑀𝑗)=𝛿𝑖𝑗, where 𝑀𝑗 is a vertex of the considered triangulation. We introduce the following discrete spaces ğ‘‰â„Ž of finite element: ğ‘‰â„Ž=î‚†ğ‘£â„Žâˆˆğ¿20,𝑇,𝐻10(Ω)∩𝐶0,𝑇,𝐻10Ω,suchthatğ‘£â„Ž|𝑘∈𝑃1,ğ‘˜âˆˆğœâ„Ž,ğ‘£â„Žâ‰¤ğ‘Ÿâ„Žğœ“,ğ‘£â„Ž(⋅,0)=ğ‘£â„Ž0.inΩ.(2.5)

We consider ğ‘Ÿâ„Ž to be the usual interpolation operator defined by 𝑣∈𝐿20,𝑇,𝐻10(Ω)∩𝐶0,𝑇,𝐻10Ω,ğ‘Ÿâ„Žğ‘£=𝑚(ℎ)𝑖=1𝑣𝑀𝑖𝜑𝑖(𝑥).(2.6)

The Discrete Maximum Principle Assumption (see [10])
The matrix whose coefficients ğ‘Ž(𝜑𝑖,𝜑𝑗) are supposed to be M-matrix. For convenience, in all the sequels, 𝐶 will be a generic constant independent on ℎ.

2.1.1. Priory Estimate

Theorem 2.3. Let us assume that the discrete bilinear form ğ‘Ž(⋅,⋅) defined as (2.2) is weakly coercive in ğ‘‰â„ŽâŠ‚ğ»10(Ω). Then, there exists two constants 𝛼>0 and 𝜆>0 such that ğ‘Žî€·ğ‘¢â„Ž,ğ‘¢â„Žî€¸â€–â€–ğ‘¢+ğœ†â„Žâ€–â€–2â€–â€–ğ‘¢â‰¥ğ›¼â„Žâ€–â€–1,(2.7) where ‖‖𝑏𝜆=𝑗‖‖2∞+𝛾2𝛾2+â€–â€–ğ‘Ž0â€–â€–âˆžîƒªğ›¾,𝛼=2.(2.8)

Proof. The bilinear form ğ‘Ž(⋅,⋅) is defined by ğ‘Žî€·ğ‘¢â„Ž,ğ‘¢â„Žî€¸=Ω𝑁𝑖𝑗=1ğ‘Žğ‘–ğ‘—(𝑥)ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘–ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘—+𝑁𝑗=1𝑏𝑗(𝑥)ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘—ğ‘¢â„Ž+ğ‘Ž0(𝑥)𝑢2â„Žîƒªğ‘‘ğ‘¥,(2.9) under assumption (1.6), we have 𝑁𝑖𝑗=1î€œÎ©ğ‘Žğ‘–ğ‘—(𝑥)ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘–ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘—>𝛾𝑁𝑖=1î‚µî€œÎ©ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘–î‚¶2‖‖=ğ›¾âˆ‡ğ‘¢â„Žâ€–â€–22,(2.10) and since |||||𝑁𝑗=1Ω𝑏𝑗(𝑥)ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘—ğ‘¢â„Ž|||||≤sup𝑗||𝑏𝑗||𝑁𝑗=1Ω||||ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘—ğ‘¢â„Ž||||â‰¤â€–â€–ğ‘ğ‘—â€–â€–âˆžâ€–â€–âˆ‡ğ‘¢â„Žâ€–â€–2â€–â€–ğ‘¢â„Žâ€–â€–2,(2.11) then we make use of the algebraic inequality 1ğ‘Žğ‘â‰¤2î€·ğ‘Ž2+𝑏2,âˆ€ğ‘Ž,𝑏∈ℝ,∀𝛾>0,(2.12) and choosing â€–â€–ğ‘Ž=âˆ‡ğ‘¢â„Žâ€–â€–2⋅√‖‖𝑏𝛾,𝑏=ğ‘—â€–â€–âˆžâˆšğ›¾â€–â€–ğ‘¢â„Žâ€–â€–2,(2.13) then we end up with |||||𝑁𝑗=1Ω𝑏𝑗(𝑥)ğœ•ğ‘¢â„Žğœ•ğ‘¥ğ‘—ğ‘¢â„Ž|||||𝛾≥−2â€–â€–âˆ‡ğ‘¢â„Žâ€–â€–22+â€–â€–ğ‘ğ‘—â€–â€–âˆžâ‹…â€–â€–ğ‘¢â„Žâ€–â€–22𝛾,(2.14) so we get ğ‘Žî€·ğ‘¢â„Ž,ğ‘¢â„Žî€¸â€–â€–â‰¥ğ›¾âˆ‡ğ‘¢â„Žâ€–â€–22−𝛾2â€–â€–âˆ‡ğ‘¢â„Žâ€–â€–22+â€–â€–ğ‘ğ‘—â€–â€–âˆžâ€–â€–ğ‘¢2ğ›¾â„Žâ€–â€–2âˆ’â€–â€–ğ‘Ž0â€–â€–âˆžâ€–â€–ğ‘¢â„Žâ€–â€–22.(2.15) It can easily verified that ğ‘Žî€·ğ‘¢â„Ž,ğ‘¢â„Žî€¸â‰¥ğ›¾2î‚€â€–â€–âˆ‡ğ‘¢â„Žâ€–â€–2+â€–â€–ğ‘¢â„Žâ€–â€–22î‚âˆ’îƒ©â€–â€–ğ‘Žğ‘—â€–â€–2∞+𝛾2𝛾2+â€–â€–ğ‘Ž0â€–â€–âˆžîƒªâ€–â€–ğ‘¢â„Žâ€–â€–22.(2.16) Consequently, we deduce from above that ğ‘Žî€·ğ‘¢â„Ž,ğ‘¢â„Žî€¸â€–â€–ğ‘¢+ğœ†â„Žâ€–â€–22â€–â€–ğ‘¢â‰¥ğ›¼â„Žâ€–â€–21𝛾suchthat𝛼=2îƒ©â€–â€–ğ‘Ž,𝜆=𝑗‖‖2∞+𝛾2𝛾2+â€–â€–ğ‘Ž0‖‖∞.(2.17)

We can identify the following result on the time energy behavior: ğ¸â„Ž(𝑡)=Ω𝑢2â„Žğ‘‘ğ‘¥.(2.18)

Setting 𝑣=0 on (2.1) and after discretization by the finite element in the ğ‘‰â„Ž, we have the semidiscretization problem î‚µğœ•ğ‘¢â„Žğœ•ğ‘¡,ğ‘¢â„Žî‚¶î€·ğ‘¢+ğ‘Žâ„Ž,ğ‘¢â„Žî€¸=12Ω𝜕𝑢2â„Žî€·ğ‘¢ğœ•ğ‘¡ğ‘‘ğ‘¥+ğ‘Žâ„Ž,ğ‘¢â„Žî€¸â‰¤î€·ğ‘“,ğ‘¢â„Žî€¸.(2.19)

Using Theorem 2.3, we deduce that12Ω𝑑𝑢2â„Žî€·ğ‘¢ğ‘‘ğ‘¡ğ‘‘ğ‘¥+ğ‘Žâ„Ž,ğ‘¢â„Žî€¸â‰¥12𝑑𝑑𝑡Ω𝑢2â„Žâ€–â€–ğ‘¢ğ‘‘ğ‘¥+ğ›¼â„Žâ€–â€–21â€–â€–ğ‘¢âˆ’ğœ†â„Žâ€–â€–22=12𝑑𝑑𝑡Ω𝑢2â„Žâ€–â€–ğ‘¢ğ‘‘ğ‘¥+ğ›¼â„Žâ€–â€–22‖‖+ğ›¼âˆ‡ğ‘¢â„Žâ€–â€–22â€–â€–ğ‘¢âˆ’ğœ†â„Žâ€–â€–22=12î‚€ğ‘‘ğ¸ğ‘‘ğ‘¡â„Ž(𝑡)+2(𝛼−𝜆)ğ¸â„Žâ€–â€–(𝑡)+2ğ›¼âˆ‡ğ‘¢â„Žâ€–â€–22≥1𝑑𝑥2î‚µğ‘‘ğ¸ğ‘‘ğ‘¡â„Ž(𝑡)+2(𝛼−𝜆)ğ¸â„Ž(𝑡)+2𝛼𝐶2∗Ω𝑢2ℎ.𝑑𝑥(2.20)

Thus, we have𝑑𝑑𝑡Ω𝑢2â„Žâ€–â€–ğ‘¢ğ‘‘ğ‘¥+ğ›¼â„Žâ€–â€–21â€–â€–ğ‘¢âˆ’ğœ†â„Žâ€–â€–22â‰¥ğ‘‘î€·ğ¸ğ‘‘ğ‘¡â„Žî€¸î‚µğ›¼(𝑡)+2𝛼−𝜆+𝐶2âˆ—î‚¶ğ¸â„Ž(𝑡).(2.21)

Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find𝑓,ğ‘¢â„Žî€¸=Ω𝑓(𝑥,𝑡)ğ‘¢â„Ž(𝑥,𝑡)𝑑𝑥≤‖𝑓‖2â€–â€–ğ‘¢â„Žâ€–â€–2.(2.22)

So that ğ‘‘ğ¸ğ‘‘ğ‘¡â„Žî‚µğ›¼(𝑡)+2𝛼−𝜆+𝐶2âˆ—î‚¶ğ¸â„Ž(𝑡)≤2‖𝑓‖2â€–â€–ğ‘¢â„Žâ€–â€–2.(2.23)

Using Young’s inequalityğ‘Žğ‘â‰¤ğœ€ğ‘Ž2+1𝑏4𝜀2,âˆ€ğ‘Ž,𝑏∈ℝ,∀𝜀>0.(2.24)

Thus, we obtainğ‘‘ğ¸ğ‘‘ğ‘¡â„Žî‚µğ›¼(𝑡)+2𝛼−𝜆+𝐶2âˆ—î‚¶ğ¸â„Ž(𝑡)≤2ğœ€ğ¸â„Ž1(𝑡)+2𝜀‖𝑓‖22,(2.25)

taking 𝜂=𝛼−𝜆+𝛼/𝐶2∗, thus we have ğ‘‘ğ¸ğ‘‘ğ‘¡â„Ž(𝑡)+2(𝜂−𝜀)ğ¸â„Ž1(𝑡)≤2𝜀‖𝑓‖22.(2.26)

Or, equivalently 𝑒2(𝜂−𝜀)ğ‘¡ğ¸â„Ž(𝑡)≤1𝑒2𝜀2(𝜂−𝜀)𝑡Ω(𝑓(𝑥,𝑡))2𝑑𝑥.(2.27)

Integrating the last inequality from 0 to 𝑡, we get ğ¸â„Ž(𝑡)≤𝑒−2(𝜂−𝜀)ğ‘¡ğ¸â„Ž(10)+2𝜀𝑡0𝑒2(𝜂−𝜀)(𝑠−𝑡)Ω(𝑓(𝑥,𝑠))2𝑑𝑥𝑑𝑠.(2.28)

Remark 2.4. In particular, when 𝑓=0 and choosing 𝜀<𝜂, then (2.28) shows that the energy 𝐸(𝑡) decreasing exponentially fast in time.

3. The 𝜃-Scheme Method for the Parabolic Variational Inequalities

3.1. Stability Analysis for the P.V.I

We apply the finite element method to approximate inequality (2.1), and the discrete P.V.I takes the form of î‚µğœ•ğ‘¢â„Žğœ•ğ‘¡,ğ‘£â„Žâˆ’ğ‘¢â„Žî‚¶î€·ğ‘¢+ğ‘Žâ„Ž,ğ‘£â„Žâˆ’ğ‘¢â„Žî€¸â‰¥î€·ğ‘“,ğ‘£â„Žâˆ’ğ‘¢â„Žî€¸,ğ‘£â„Žâˆˆğ‘‰â„Ž.(3.1)

Now, we apply the 𝜃-scheme on the semidiscrete problem (3.1); for any 𝜃∈[0,1] and 𝑘=1,…,𝑛, we have î€·ğ‘¢ğ‘˜â„Žâˆ’ğ‘¢â„Žğ‘˜âˆ’1,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘𝑢+Î”ğ‘¡â‹…ğ‘Žâ„Žğœƒ,𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘𝑓≥Δ𝑡⋅𝜃,𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘,ğ‘£â„Žâˆˆğ‘‰â„Ž,(3.2) where ğ‘¢â„Žğœƒ,𝑘=ğœƒğ‘¢ğ‘˜â„Ž+(1−𝜃)ğ‘¢â„Žğ‘˜âˆ’1,𝑓𝜃,𝑘𝑡=𝜃𝑓𝑘𝑡+(1−𝜃)𝑓𝑘−1.(3.3)

It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear form ğ‘Ž(⋅,⋅). We recall that 𝑊 is compactly embedded in 𝐿2(Ω), since Ω is bounded. Thus, there exists a nondecreasing sequence of eigenvalues 𝛿≤𝜆1≤𝜆2≤⋯ for the bilinear form ğ‘Ž(⋅,⋅) satisfying 𝜔𝑗∈𝐿2,𝜔𝑗𝜔≠0:ğ‘Žğ‘—,ğ‘£â„Žî€¸=𝜆𝑗𝜔𝑗,ğ‘£â„Žî€¸,âˆ€ğ‘£â„Žâˆˆğ‘‰â„Ž.(3.4) The corresponding eigenfunctions {𝜔𝑗} form a complete orthonormal basis in 𝐿2(Ω). In analogous way, when considering the finite dimensional problem in ğ‘Šâ„Ž, we find a sequence of eigenvalues 𝛿≤𝜆1â„Žâ‰¤ğœ†2â„Žâ‰¤â‹¯â‰¤ğœ†ğ‘š(ℎ) and 𝐿2-orthonormal basis of eigenvectorss ğœ”ğ‘–â„Žâˆˆğ‘Šâ„Ž,𝑖=1,2,…,𝑚(ℎ). Any function ğ‘£â„Ž in ğ‘‰â„Ž can thus be expanded with respect to the system ğœ”ğ‘–â„Ž as ğ‘£â„Ž=𝑚(ℎ)𝑖=1î€·ğ‘£â„Ž,ğœ”ğ‘–â„Žî€¸ğœ”ğ‘–â„Ž,(3.5) in particular, we have ğ‘¢ğ‘˜â„Ž=𝑚(ℎ)𝑖=1ğ‘¢ğ‘˜ğ‘–ğœ”ğ‘–â„Ž,𝑢𝑘𝑖=î€·ğ‘¢ğ‘˜â„Ž,ğœ”ğ‘–â„Žî€¸.(3.6) Moreover, let ğ‘“ğ‘˜â„Ž be the 𝐿2-orthogonal projection of 𝑓𝜃,𝑘 into ğ‘Šâ„Ž, that is, ğ‘“ğ‘˜â„Žâˆˆğ‘Šâ„Ž and î€·ğ‘“ğ‘˜â„Ž,ğ‘£â„Žî€¸=𝑓𝜃,𝑘,ğ‘£â„Žî€¸,(3.7) and set ğ‘“ğ‘˜â„Ž=𝑚(ℎ)𝑖=1ğ‘“ğ‘˜ğ‘–ğœ”ğ‘–â„Ž,𝑓𝑘𝑖=î€·ğ‘“ğ‘˜â„Ž,ğœ”ğ‘–â„Žî€¸.(3.8) We are now in a position to prove the stability for 𝜃∈[0,1/2[

Choosing in (3.1) ğ‘£â„Ž=0, thus we have 1î€·ğ‘¢Î”ğ‘¡ğ‘˜â„Žâˆ’ğ‘¢â„Žğ‘˜âˆ’1,ğ‘¢â„Žğœƒ,𝑘𝑢+ğ‘Žâ„Žğœƒ,𝑘,ğ‘¢â„Žğœƒ,𝑘≤𝑓𝜃,𝑘,ğ‘¢â„Žğœƒ,𝑘,ğ‘¢â„Žğœƒ,ğ‘˜âˆˆğ‘‰â„Ž.(3.9)

The inequalities (3.2) is equivalent to 1𝑢Δ𝑡𝑘𝑖−𝑢𝑖𝑘−1+ğœ†ğ‘–â„Žî€·ğœƒğ‘¢ğ‘˜ğ‘–+(1−𝜃)𝑢𝑖𝑘−1≤𝑓𝑘𝑖.(3.10)

Since ğœ”ğ‘–â„Ž are the eigenfunctions means ğ‘Žî€·ğœ”ğ‘–â„Ž,ğœ”ğ‘–â„Žî€¸=ğœ†ğ‘–â„Žî€·ğœ”ğ‘–â„Ž,ğœ”ğ‘–â„Žî€¸=ğœ†ğ‘–â„Žâ‹…ğ›¿ğ‘–ğ‘–=ğœ†ğ‘–â„Ž,(3.11) for each 𝑘=0,…,𝑚(ℎ)−1, we can rewrite (3.9) as 𝑢𝑘𝑖≤1−(1−𝜃)â‹…Î”ğ‘¡â‹…ğœ†ğ‘–â„Ž1+ğœƒÎ”ğ‘¡â‹…ğœ†ğ‘–â„Žğ‘¢ğ‘–ğ‘˜âˆ’1+Δ𝑡1+ğœƒÎ”ğ‘¡â‹…ğœ†ğ‘–â„Žğ‘“ğ‘˜ğ‘–,(3.12) this inequality system stable if and only if ||||1−(1−𝜃)â‹…Î”ğ‘¡â‹…ğœ†ğ‘–â„Ž1+ğœƒÎ”ğ‘¡â‹…ğœ†ğ‘–â„Ž||||<1,(3.13) that is to say 22𝜃−1>âˆ’ğœ†ğ‘–â„Žâ‹…Î”ğ‘¡(3.14) means 2Δ𝑡<(1−2𝜃)ğœ†ğ‘–â„Ž.(3.15)

So that this relation satisfied for all the eigenvalues ğœ†ğ‘–â„Ž of bilinear form ğ‘Ž(⋅,⋅), we have to choose their highest value, and we take it for ğœ†ğ‘šâ„Ž=𝜌(𝐴) (rayon spectral)

We deduce that if 𝜃≥1/2 the 𝜃-scheme way is stable unconditionally (i.e., stable for all Δt). However, if 0≤𝜃<1/2 the 𝜃-scheme is stable unless 2Δ𝑡<(1−2𝜃)𝜌(𝐴).(3.16)

We can prove that there exist two positive constants 𝑐1,𝑐2 such that 𝑐1ℎ2â‰¤ğœ†ğ‘šâ„Ž=𝑐2ℎ−2,(3.17) thus the method of 𝜃-scheme is stable if and only if Δ𝑡<2ğ¶â„Ž(1−2𝜃)2.(3.18)

Notice that this condition is always satisfied if 0≤𝜃<1/2. Hence, taking the absolute value of (3.12), we have ||𝑢𝑚𝑖||<||𝑢0𝑖||+Δ𝑡1+ğœƒÎ”ğ‘¡â‹…ğœ†ğ‘–â„Žğ‘šâˆ’1𝑖=1𝑓𝑘𝑖,(3.19) also we deduce that â€–â€–ğ‘¢ğ‘šğ‘–â€–â€–âˆž<‖‖𝑢0ğ‘–â€–â€–âˆž+‖‖‖Δ𝑡1+ğœƒÎ”ğ‘¡â‹…ğœ†ğ‘–â„Žâ€–â€–â€–âˆžğ‘šâˆ’1𝑖=1â€–â€–ğ‘“ğ‘˜ğ‘–â€–â€–âˆž.(3.20)

Remark 3.1 (cf. [4]). We assume that the coerciveness assumption (Theorem 2.3) is satisfied with 𝜆=0, and for each 𝑘=1,…,𝑛, we find â€–â€–ğ‘¢ğ‘˜â„Žâ€–â€–22+2Δ𝑡𝑛𝑘=1ğ‘Žî€·ğ‘¢â„Žğœƒ,𝑘,ğ‘¢â„Žğœƒ,𝑘≤𝐶(𝑛)𝑛𝑘=1‖‖𝑓Δ𝑡𝜃,𝑘‖‖22.(3.21)

4. Asymptotic Behavior of 𝜃-Scheme for the P.V.I

This section is devoted to the proof of the main result of the present paper; we need first to study some properties such as proving the existence and uniqueness for parabolic variational inequalities.

4.1. Existence and Uniqueness for P.V.I

Theorem 4.1 (cf. [2, 3]). Under the previous assumptions, and the maximum principle, there exists a constant 𝐶 independent of ℎ such that â€–â€–ğ‘¢âˆžâˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤ğ¶â„Ž2||||logℎ2,(4.1) where ğ‘¢âˆž and ğ‘¢âˆžâ„Ž are, respectively, stationery solutions to the following continue and discrete inequalities: 𝑏(ğ‘¢âˆž,ğ‘£âˆ’ğ‘¢âˆž)≥(𝑓+ğœ†ğ‘¢âˆž,ğ‘£âˆ’ğ‘¢âˆž),𝑣∈𝐻10𝑏𝑢(Ω),(4.2)∞ℎ,ğ‘£â„Žâˆ’ğ‘¢âˆžâ„Žî€¸â‰¥î€·ğ‘“+ğœ†ğ‘¢âˆž,ğ‘£â„Žâˆ’ğ‘¢âˆžâ„Žî€¸,ğ‘£â„Žâˆˆğ‘‰â„Ž,(4.3) such that 𝑏(⋅,⋅)=ğ‘Ž(⋅,⋅)+𝜆(⋅,⋅),(4.4)where 𝜆 is a positive constant arbitrary.We have ğ‘¢â„Žğœƒ,𝑘=ğœƒğ‘¢ğ‘˜â„Ž+(1−𝜃)ğ‘¢â„Žğ‘˜âˆ’1â‰¤ğœƒğ‘Ÿâ„Žğœ“+(1−𝜃)ğ‘Ÿâ„Žğœ“=ğ‘Ÿâ„Žğœ“.Thus, we can rewrite (3.1) as, for ğ‘¢â„Žğœƒ,ğ‘˜âˆˆğ‘‰â„Žîƒ©ğ‘¢â„Žğœƒ,𝑘𝜃Δ𝑡,ğ‘£â„Žâˆ’Ìƒğ‘¢ğ‘˜â„Žîƒªî€·ğ‘¢+ğ‘Žâ„Žğœƒ,𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘≥𝑓𝜃,𝑘+ğ‘¢â„Žğ‘˜âˆ’1𝜃Δ𝑡,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘,ğ‘£â„Žâˆˆğ‘‰â„Ž.(4.5) Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities: ğ‘î€·ğ‘¢â„Žğœƒ,𝑘,ğ‘£â„Žâˆ’Ìƒğ‘¢ğ‘˜â„Žî€¸â‰¥î€·ğ‘“ğœƒ,𝑘+ğœ‡ğ‘¢â„Žğ‘˜âˆ’1,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘,ğ‘£â„Žâˆˆğ‘‰â„Ž,(4.6) such that ğ‘î€·ğ‘¢â„Žğœƒ,𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘𝑢=ğœ‡â„Žğœƒ,𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘𝑢+ğ‘Žâ„Žğœƒ,𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,𝑘,ğ‘£â„Ž,ğ‘¢â„Žğœƒ,ğ‘˜âˆˆğ‘‰â„Ž,1𝜇==𝑇𝜃Δ𝑡,𝜃𝑘(4.7) where ğ‘¢â„Žğœƒ,1 is the solution to the following discrete inequality: ğ‘Žî€·ğ‘¢â„Žğœƒ,1,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,1=𝑔𝑡𝑘,ğ‘£â„Žâˆ’ğ‘¢â„Žğœƒ,1,ğ‘£â„Žâˆˆğ‘‰â„Ž,(4.8) where 𝑔(𝑡𝑘) is a regular function given.

4.1.1. A Fixed Point Mapping Associated with Discrete Problem (4.7)

We consider the mapping ğ‘‡â„Žâˆ¶ğ¿âˆž+(Ω)âŸ¶ğ‘‰â„Žğ‘¤âŸ¶ğ‘‡â„Ž(𝑤)=ğœ‰â„Ž,(4.9) where ğœ‰â„Ž is the unique solution of the following P.V.I: find ğœ‰â„Žâˆˆğ‘‰â„Žğ‘î€·ğœ‰â„Ž,ğ‘£â„Žâˆ’ğœ‰â„Žî€¸â‰¥î€·ğ‘“ğœƒ,𝑘+𝜇𝑤,ğ‘£â„Žâˆ’ğœ‰â„Žî€¸,ğ‘£â„Žâˆˆğ‘‰â„Ž.(4.10)

Proposition 4.2. Under the previous hypotheses and notations, if one sets 𝜃≥1/2, the mapping ğ‘‡â„Ž is a contraction in ğ¿âˆž(Ω) with rate of contraction 1/(1+𝛽⋅𝜃⋅Δ𝑡). Therefore, ğ‘‡â„Ž admits a unique fixed point which coincides with the solution of P.V. I (4.7).

Proof. For 𝑤, 𝑤 in ğ¿âˆž(Ω), we consider ğœ‰â„Ž=ğ‘‡â„Ž(𝑤)=𝜕(𝑓𝜃,𝑘+𝜇𝑤,ğ‘Ÿâ„Žğœ“) and Ìƒğœ‰â„Ž=ğ‘‡â„Ž(𝑤)=𝜕(𝑓𝜃,𝑘+𝜇𝑤,ğ‘Ÿâ„Žî‚ğœ“) solution to quasivariational inequalities (4.7) with right-hand side 𝐹𝜃,𝑘=𝑓𝜃,𝑘+ğœ‡ğ‘¤â„Ž,𝐹𝜃,𝑘=𝑓𝜃,𝑘𝑤+ğœ‡â„Ž.
Now, setting 1𝜙=‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,ğ‘˜â€–â€–âˆž,(4.11) then for ğœ‰â„Ž+𝜙 is solution of ğ‘î€·ğœ‰â„Žî€·ğ‘£+𝜙,â„Žî€¸âˆ’î€·ğœ‰+ğœ™â„Žâ‰¥î€·ğ¹+𝜙𝜃,𝑘+ğ‘Ž0𝑣𝜙,â„Žî€¸âˆ’î€·ğœ‰+ğœ™â„Ž,+𝜙𝜉+ğœ™â‰¤ğ‘Ÿâ„Žğœ“+𝜙,ğ‘£â„Ž+ğœ™â‰¤ğ‘Ÿâ„Žğœ“+𝜙,âˆ€ğ‘£â„Žâˆˆğ‘‰â„Ž.(4.12) Also, we have 𝐹𝜃,𝑘≤𝐹𝜃,𝑘+‖‖𝐹𝜃,𝑘−𝐹𝜃,ğ‘˜â€–â€–âˆžâ‰¤î‚ğ¹ğœƒ,𝑘+ğ‘Ž0‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,ğ‘˜â€–â€–âˆžâ‰¤î‚ğ¹ğœƒ,𝑘+ğ‘Ž0𝜙,(4.13) thus ğœ•â„Žî€·ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žî€¸ğœ“+ğœ™â‰¤ğœ•â„Žî‚€î‚ğ¹ğœƒ,𝑘+ğ‘Ž0(𝑥)𝜙,ğ‘Ÿâ„Žî‚î‚ğœ“+ğœ™â‰¤ğœ•â„Žî‚€î‚ğ¹ğœƒ,𝑘,𝑟𝜓+𝜙,(4.14) hence ğœ‰â„Žâ‰¤Ìƒğœ‰â„Ž+𝜙.(4.15) Similarly, interchanging the roles of 𝑤 and 𝑤, we also get Ìƒğœ‰â„Žâ‰¤ğœ‰â„Ž+𝜙.(4.16) Finally, this yields â€–â€–ğœ•â„Žî€·ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žğœ“î€¸âˆ’ğœ•â„Žî‚€î‚ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žî‚â€–â€–î‚ğœ“âˆžâ‰¤1‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,ğ‘˜â€–â€–âˆžâ‰¤1‖‖𝑓𝛽+𝜇𝜃,𝑘+𝜇𝑤−𝑓𝜃,ğ‘˜î‚ğ‘¤â€–â€–âˆ’ğœ‡âˆžâ€–â€–î‚ğ‘¤â€–â€–ğ‘¤âˆ’âˆžâ‰¤1‖‖𝑤‖‖1+ğ›½ğœƒÎ”ğ‘¡ğ‘¤âˆ’âˆž,(4.17) which completes the proof.

Remark 4.3. If we set 0≤𝜃<1/2, the mapping ğ‘‡â„Ž is a contraction in ğ¿âˆž(Ω) with rate of contraction 2/(2+𝛽𝜃(1−2𝜃)𝜌(𝐴)), where 𝜌(𝐴) is a spectral radius of operator 𝐴.

Proof. Under condition of stability, we have shown the 𝜃-scheme is stable if and only if Δ𝑡<(2𝐶/(1−2𝜃))ℎ2.
Thus it can be easily show that â€–â€–ğœ•â„Žî€·ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žğœ“î€¸âˆ’ğœ•â„Žî‚€î‚ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žî‚â€–â€–î‚ğœ“âˆžâ‰¤1‖‖𝑤‖‖1+ğ›½ğœƒÎ”ğ‘¡ğ‘¤âˆ’âˆžâ‰¤2‖‖𝑤‖‖2+𝛽𝜃(1−2𝜃)𝜌(𝐴)ğ‘¤âˆ’âˆžâ‰¤11+𝛽𝜃(1−2𝜃)/2ğ¶â„Ž2î€¸â€–â€–î‚ğ‘¤â€–â€–ğ‘¤âˆ’âˆž=2ğ¶â„Ž22ğ¶â„Ž2‖‖𝑤‖‖+𝛽𝜃(1−2𝜃)ğ‘¤âˆ’âˆž,(4.18) also it can be found that â€–â€–ğœ•â„Žî€·ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žğœ“î€¸âˆ’ğœ•â„Žî‚€î‚ğ¹ğœƒ,𝑘,ğ‘Ÿâ„Žî‚â€–â€–î‚ğœ“âˆžâ‰¤1(1+𝛽𝜃1−2𝜃)/2ğ¶â„Ž2î€¸â€–â€–î‚ğ‘¤â€–â€–ğ‘¤âˆ’âˆž=2ğ¶â„Ž22ğ¶â„Ž2‖‖𝑤‖‖+𝛽𝜃(1−2𝜃)ğ‘¤âˆ’âˆž,(4.19) thus the mapping ğ‘‡â„Ž is a contraction in ğ¿âˆž(Ω) with rate of contraction (2ğ¶â„Ž2)/(2ğ¶â„Ž2+𝛽𝜃(1−2𝜃)). Therefore, ğ‘‡â„Ž admits a unique fixed point which coincides with the solution of P.V.I (4.7) â€–â€–ğ‘‡â„Ž(𝑤)−Tâ„Žî€·î‚ğ‘¤î€¸â€–â€–âˆžâ‰¤1‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,ğ‘˜â€–â€–âˆž=1‖‖𝑓𝛽+𝜇𝜃,𝑘+𝜇𝑤−𝑓𝜃,ğ‘˜î‚ğ‘¤â€–â€–âˆ’ğœ‡âˆžâ‰¤ğœ‡â€–â€–î‚ğ‘¤â€–â€–ğ›½+ğœ‡ğ‘¤âˆ’âˆžâ‰¤1‖‖𝑤‖‖1+ğ›½ğœƒÎ”ğ‘¡ğ‘¤âˆ’âˆž.(4.20) This completes the proof.

4.2. Discrete Algorithm

Starting from 𝑢0ℎ=𝑢0ℎ (initial data) and the ğ‘¢â„Žğœƒ,1 solution of problem (4.7), we introduce the following discrete algorithm: ğ‘¢â„Žğœƒ,𝑘=ğ‘‡â„Žğ‘¢â„Žğ‘˜âˆ’1,𝑘=1,…,𝑛,(4.21) where ğ‘¢â„Žğœƒ,𝑘 is the solution of the problem (4.7).

Remark 4.4. If we choose 𝜃=1 in (4.21), we get Bensoussan’s algorithm. The idea of this choice has been studied by Boulbrachen (cf. [3]).

Proposition 4.5. Under the previous hypotheses, one has the following estimate of convergence: if 𝜃≥1/2â€–â€–ğ‘¢â„Žğœƒ,ğ‘˜âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ11+ğ›½ğœƒÎ”ğ‘¡ğ‘˜â€–â€–ğ‘¢âˆžâ„Žâˆ’ğ‘¢â„Ž0‖‖∞,(4.22) and one has for â€–â€–ğ‘¢â„Žğœƒ,ğ‘˜âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ2ğ¶â„Ž22ğ¶â„Ž2+𝛽𝜃(1−2𝜃)ğ‘˜â€–â€–ğ‘¢âˆžâ„Žâˆ’ğ‘¢â„Ž0‖‖∞1for0≤𝜃<2.(4.23)

Proof. we set a first case 𝜃≥1/2, and we have ğ‘¢âˆžâ„Ž=ğ‘‡â„Žğ‘¢âˆžâ„Ž,â€–â€–ğ‘¢â„Žğœƒ,1âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž=â€–â€–ğ‘‡â„Žğ‘¢0â„Žâˆ’ğ‘‡â„Žğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ1‖‖𝑢1+𝛽𝜃Δ𝑡0â„Žâˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž.(4.24) for 𝑘≥2, we use the Bensoussan-Lions' algorithm (ğ‘¢ğ‘˜â„Ž=ğ‘‡â„Žğ‘¢â„Žğ‘˜âˆ’1,𝑘=1,…,𝑛) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that â€–â€–ğ‘¢â„Žğœƒ,ğ‘˜âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ11+𝛽𝜃Δ𝑡𝑘‖‖𝑢0â„Žâˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž,(4.25) so â€–â€–ğ‘¢â„Žğœƒ,𝑘+1âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž=â€–â€–ğ‘‡â„Žğ‘¢ğ‘˜â„Žâˆ’ğ‘‡â„Žğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ1‖‖𝑢1+ğ›½ğœƒÎ”ğ‘¡ğ‘˜â„Žâˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž,(4.26) thus â€–â€–ğ‘¢â„Žğœƒ,𝑘+1âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ11+𝛽𝜃Δ𝑡𝑘+1‖‖𝑢0â„Žâˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž,(4.27) for a second case 0≤𝜃<1/2, it can be easily shown that â€–â€–ğ‘¢â„Žğœƒ,ğ‘˜âˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆžâ‰¤î‚µ2ğ¶â„Ž22ğ¶â„Ž2+𝛽𝜃(1−2𝜃)𝑘‖‖𝑢0â„Žâˆ’ğ‘¢âˆžâ„Žâ€–â€–âˆž.(4.28)

4.2.1. Asymptotic Behavior

This section is devoted to the proof of main result of the present paper, where we prove the theorem of the asymptotic behavior in ğ¿âˆž-norm for parabolic variational inequalities

Now, we evaluate the variation in ğ¿âˆž between ğ‘¢ğœƒâ„Ž(𝑇,𝑥), the discrete solution calculated at the moment 𝑇=𝑛Δ𝑡 and ğ‘¢âˆž, the asymptotic continuous solution of (4.2)

Theorem 4.6 (The main result). Under condition of Theorem 4.1 and Proposition 4.5, one has for the first case 𝜃≥1/2, â€–â€–ğ‘¢â„Žğœƒ,ğ‘›âˆ’ğ‘¢âˆžâ€–â€–âˆžî‚¸â„Žâ‰¤ğ¶2||||logℎ2+11+𝛽𝜃Δ𝑡𝑛,(4.29) and for the second case 0≤𝜃<1/2, â€–â€–ğ‘¢â„Žğœƒ,ğ‘›âˆ’ğ‘¢âˆžâ€–â€–âˆžî‚¸â„Žâ‰¤ğ¶2||||logℎ2+2ğ¶â„Ž22ğ¶â„Ž2+𝛽𝜃(1−2𝜃)𝑛,(4.30) where 𝐶 is a constant independent of ℎ and 𝑘.

Proof. We have ğ‘¢â„Žğœƒ,𝑘(𝑥)=ğ‘¢â„Ž][(𝑡,𝑥)for𝑡∈(𝑘−1)Δ𝑡;𝑘Δ𝑡,(4.31) thus ğ‘¢â„Žğœƒ,𝑛(𝑥)=ğ‘¢â„Ž(𝑇,𝑥),(4.32) then â€–â€–ğ‘¢ğœƒâ„Ž(𝑇,𝑥)âˆ’ğ‘¢âˆžâ€–â€–ğ¿âˆž(Ω)=â€–â€–ğ‘¢â„Žğœƒ,ğ‘›âˆ’ğ‘¢âˆžâ€–â€–ğ¿âˆž(Ω)â‰¤â€–â€–ğ‘¢â„Žğœƒ,ğ‘›âˆ’ğ‘¢âˆžâ„Žâ€–â€–ğ¿âˆž(Ω)+â€–â€–ğ‘¢âˆžâ„Žâˆ’ğ‘¢âˆžâ€–â€–ğ¿âˆž(Ω).(4.33) Using, Theorem 4.1 and Proposition 4.5, we have for 𝜃≥1/2, â€–â€–ğ‘¢â„Žğœƒ,ğ‘›âˆ’ğ‘¢âˆžâ€–â€–âˆžî‚¸â„Žâ‰¤ğ¶2||||logℎ2+11+𝛽𝜃Δ𝑡𝑛,(4.34) and for 0≤𝜃<1/2, we have â€–â€–ğ‘¢â„Žğœƒ,ğ‘›âˆ’ğ‘¢âˆžâ€–â€–âˆžî‚¸â„Žâ‰¤ğ¶2||||logℎ2+2ğ¶â„Ž22ğ¶â„Ž2+𝛽𝜃(1−2𝜃)𝑛.(4.35)

5. Conclusion

In this paper, we have introduced a new approach for the theta time scheme combined with a finite element spatial approximation of parabolic variational inequalities (P.V.I). We have given a simple result to time energy behavior and established a convergence and asymptotic behavior in uniform norm. The type of estimation, which we have obtained here, is important for the calculus of quasistationary state for the simulation of petroleum or gaseous deposit. A future paper will be devoted to the computation of this method, where efficient numerical monotone algorithms will be treated.

Acknowledgment

The authors would like to thank the referee and the editors for reading and suggestions.

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Copyright © 2011 Salah Boulaaras and Mohamed Haiour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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