Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 703670, 15 pages
http://dx.doi.org/10.5402/2011/703670
Research Article

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

1Hydrometeorological Institute of Formation and Research, Box. 7019, Seddikia, Oran 31000, Algeria
2Department of Mathematics, Faculty of Science, University of Annaba, Box. 12, Annaba 23000, Algeria

Received 30 March 2011; Accepted 13 May 2011

Academic Editors: G. Garcea and S. Zhang

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.

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., [18]).

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)𝐿20,𝑇,𝐻10(Ω),𝑣(𝑡,𝑥)𝜓(𝑡,𝑥),𝑣(0,𝑥)=𝑣0inΩ,(1.1) with 𝜓𝐿20,𝑇,𝑊2,(Ω).(1.2)

We consider the following problem, find 𝑢𝐾 solution of 𝜕𝑢𝜕𝑡+𝐴𝑢𝑓inΣ,𝑢(𝑡,𝑥)=0inΓ,(1.3) where Σ is a set in ×𝑁 defined as Σ=Ω×[0,𝑇] with 𝑇<+, and Ω is convex domain in 𝑁, 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=𝑁𝑖𝑗=1𝜕𝜕𝑥𝑖𝑎𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗+𝑁𝑗=1𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗+𝑎0(𝑥)𝑢,(1.4) and whose coefficients: 𝑎𝑖,𝑗(𝑥),𝑏𝑗(𝑥),𝑎0(𝑥)𝐿(Ω)𝐶2(Ω),𝑥Ω,1𝑖,𝑗𝑁 are sufficiently smooth functions and satisfy the following conditions: 𝑎𝑖𝑗(𝑥)=𝑎𝑗𝑖(𝑥);𝑎0(𝑥)𝛽>0,𝛽isaconstant,(1.5)𝑁𝑖𝑗=1𝑎𝑖𝑗(𝑥)𝜉𝑖𝜉𝑗||𝜉||𝛾2;𝜉𝑁,𝛾>0,𝑥Ω,(1.6)𝑓 is a regular functions satisfying 𝑓𝐿2(0,𝑇,𝐿(Ω))𝐶10,𝑇,𝐻1(Ω),𝑓0.(1.7)

We specify the following notations: 𝐿2(Ω)=2,1=𝐻10(Ω),𝐿(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 𝑉𝐻10. In the second step, we discretize the problem with respect to time using the 𝜃-scheme. Therefore, we search a sequence of elements 𝑢𝑛𝑉 which approaches 𝑢𝑛(𝑡𝑛),𝑡𝑛=𝑛Δ𝑡, with initial data 𝑢0=𝑢0. 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 𝐿-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,𝑣𝑢+𝑎(𝑢,𝑣𝑢)(𝑓,𝑣𝑢),𝑣𝐾,(2.1) where 𝑎(,) is the bilinear form associated with operator 𝐴 defined in (1.4). Namely, 𝑎(𝑢,𝑣)=Ω𝑁𝑖𝑗=1𝑎𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥𝑖𝜕𝑣𝜕𝑥𝑗+𝑁𝑗=1𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗𝑣+𝑎0(𝑥)𝑢𝑣𝑑𝑥,(2.2)

Theorem 2.1 (see [9]). The problem (1.3) has an unique solution 𝑢𝐾(𝑢). Moreover, one has 𝑢𝐿20,𝑇;𝐻10,(Ω)𝜕𝑢𝜕𝑡𝐿20,𝑇;𝐻1(Ω).(2.3)

Lemma 2.2 (Sobolev-Poincare inequality). Let Ω be a bounded overt in 𝑁, with sufficiently smooth boundary Γ, then there exists a 𝐶 such that 𝑢2𝐶𝑢2,𝑣𝐻10(Ω)𝐶2Ω,=𝑁𝑖=1𝜕𝜕𝑥𝑖.(2.4)

2.1. The Discrete Problem

Let us assume that Ω can be decomposed into triangles and 𝜏 denotes the set of all the elements >0, where is the mesh size. We assume that the family 𝜏 is regular and quasi-uniform, and we consider the usual basis of affine functions 𝜑𝑖, 𝑖={1,,𝑚()} defined by 𝜑𝑖(𝑀𝑗)=𝛿𝑖𝑗, where 𝑀𝑗 is a vertex of the considered triangulation. We introduce the following discrete spaces 𝑉 of finite element: 𝑉=𝑣𝐿20,𝑇,𝐻10(Ω)𝐶0,𝑇,𝐻10Ω,suchthat𝑣|𝑘𝑃1,𝑘𝜏,𝑣𝑟𝜓,𝑣(,0)=𝑣0.inΩ.(2.5)

We consider 𝑟 to be the usual interpolation operator defined by 𝑣𝐿20,𝑇,𝐻10(Ω)𝐶0,𝑇,𝐻10Ω,𝑟𝑣=𝑚()𝑖=1𝑣𝑀𝑖𝜑𝑖(𝑥).(2.6)

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

2.1.1. Priory Estimate

Theorem 2.3. Let us assume that the discrete bilinear form 𝑎(,) defined as (2.2) is weakly coercive in 𝑉𝐻10(Ω). Then, there exists two constants 𝛼>0 and 𝜆>0 such that 𝑎𝑢,𝑢𝑢+𝜆2𝑢𝛼1,(2.7) where 𝑏𝜆=𝑗2+𝛾2𝛾2+𝑎0𝛾,𝛼=2.(2.8)

Proof. The bilinear form 𝑎(,) is defined by 𝑎𝑢,𝑢=Ω𝑁𝑖𝑗=1𝑎𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥𝑖𝜕𝑢𝜕𝑥𝑗+𝑁𝑗=1𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗𝑢+𝑎0(𝑥)𝑢2𝑑𝑥,(2.9) under assumption (1.6), we have 𝑁𝑖𝑗=1Ω𝑎𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥𝑖𝜕𝑢𝜕𝑥𝑗>𝛾𝑁𝑖=1Ω𝜕𝑢𝜕𝑥𝑖2=𝛾𝑢22,(2.10) and since |||||𝑁𝑗=1Ω𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗𝑢|||||sup𝑗||𝑏𝑗||𝑁𝑗=1Ω||||𝜕𝑢𝜕𝑥𝑗𝑢||||𝑏𝑗𝑢2𝑢2,(2.11) then we make use of the algebraic inequality 1𝑎𝑏2𝑎2+𝑏2,𝑎,𝑏,𝛾>0,(2.12) and choosing 𝑎=𝑢2𝑏𝛾,𝑏=𝑗𝛾𝑢2,(2.13) then we end up with |||||𝑁𝑗=1Ω𝑏𝑗(𝑥)𝜕𝑢𝜕𝑥𝑗𝑢|||||𝛾2𝑢22+𝑏𝑗𝑢22𝛾,(2.14) so we get 𝑎𝑢,𝑢𝛾𝑢22𝛾2𝑢22+𝑏𝑗𝑢2𝛾2𝑎0𝑢22.(2.15) It can easily verified that 𝑎𝑢,𝑢𝛾2𝑢2+𝑢22𝑎𝑗2+𝛾2𝛾2+𝑎0𝑢22.(2.16) Consequently, we deduce from above that 𝑎𝑢,𝑢𝑢+𝜆22𝑢𝛼21𝛾suchthat𝛼=2𝑎,𝜆=𝑗2+𝛾2𝛾2+𝑎0.(2.17)

We can identify the following result on the time energy behavior: 𝐸(𝑡)=Ω𝑢2𝑑𝑥.(2.18)

Setting 𝑣=0 on (2.1) and after discretization by the finite element in the 𝑉, we have the semidiscretization problem 𝜕𝑢𝜕𝑡,𝑢𝑢+𝑎,𝑢=12Ω𝜕𝑢2𝑢𝜕𝑡𝑑𝑥+𝑎,𝑢𝑓,𝑢.(2.19)

Using Theorem 2.3, we deduce that12Ω𝑑𝑢2𝑢𝑑𝑡𝑑𝑥+𝑎,𝑢12𝑑𝑑𝑡Ω𝑢2𝑢𝑑𝑥+𝛼21𝑢𝜆22=12𝑑𝑑𝑡Ω𝑢2𝑢𝑑𝑥+𝛼22+𝛼𝑢22𝑢𝜆22=12𝑑𝐸𝑑𝑡(𝑡)+2(𝛼𝜆)𝐸(𝑡)+2𝛼𝑢221𝑑𝑥2𝑑𝐸𝑑𝑡(𝑡)+2(𝛼𝜆)𝐸(𝑡)+2𝛼𝐶2Ω𝑢2.𝑑𝑥(2.20)

Thus, we have𝑑𝑑𝑡Ω𝑢2𝑢𝑑𝑥+𝛼21𝑢𝜆22𝑑𝐸𝑑𝑡𝛼(𝑡)+2𝛼𝜆+𝐶2𝐸(𝑡).(2.21)

Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find𝑓,𝑢=Ω𝑓(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑𝑥𝑓2𝑢2.(2.22)

So that 𝑑𝐸𝑑𝑡𝛼(𝑡)+2𝛼𝜆+𝐶2𝐸(𝑡)2𝑓2𝑢2.(2.23)

Using Young’s inequality𝑎𝑏𝜀𝑎2+1𝑏4𝜀2,𝑎,𝑏,𝜀>0.(2.24)

Thus, we obtain𝑑𝐸𝑑𝑡𝛼(𝑡)+2𝛼𝜆+𝐶2𝐸(𝑡)2𝜀𝐸1(𝑡)+2𝜀𝑓22,(2.25)

taking 𝜂=𝛼𝜆+𝛼/𝐶2, thus we have 𝑑𝐸𝑑𝑡(𝑡)+2(𝜂𝜀)𝐸1(𝑡)2𝜀𝑓22.(2.26)

Or, equivalently 𝑒2(𝜂𝜀)𝑡𝐸(𝑡)1𝑒2𝜀2(𝜂𝜀)𝑡Ω(𝑓(𝑥,𝑡))2𝑑𝑥.(2.27)

Integrating the last inequality from 0 to 𝑡, we get 𝐸(𝑡)𝑒2(𝜂𝜀)𝑡𝐸(10)+2𝜀𝑡0𝑒2(𝜂𝜀)(𝑠𝑡)Ω(𝑓(𝑥,𝑠))2𝑑𝑥𝑑𝑠.(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 𝜕𝑢𝜕𝑡,𝑣𝑢𝑢+𝑎,𝑣𝑢𝑓,𝑣𝑢,𝑣𝑉.(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 𝑢𝑘𝑢𝑘1,𝑣𝑢𝜃,𝑘𝑢+Δ𝑡𝑎𝜃,𝑘,𝑣𝑢𝜃,𝑘𝑓Δ𝑡𝜃,𝑘,𝑣𝑢𝜃,𝑘,𝑣𝑉,(3.2) where 𝑢𝜃,𝑘=𝜃𝑢𝑘+(1𝜃)𝑢𝑘1,𝑓𝜃,𝑘𝑡=𝜃𝑓𝑘𝑡+(1𝜃)𝑓𝑘1.(3.3)

It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear form 𝑎(,). We recall that 𝑊 is compactly embedded in 𝐿2(Ω), since Ω is bounded. Thus, there exists a nondecreasing sequence of eigenvalues 𝛿𝜆1𝜆2 for the bilinear form 𝑎(,) satisfying 𝜔𝑗𝐿2,𝜔𝑗𝜔0:𝑎𝑗,𝑣=𝜆𝑗𝜔𝑗,𝑣,𝑣𝑉.(3.4) The corresponding eigenfunctions {𝜔𝑗} form a complete orthonormal basis in 𝐿2(Ω). In analogous way, when considering the finite dimensional problem in 𝑊, we find a sequence of eigenvalues 𝛿𝜆1𝜆2𝜆𝑚() and 𝐿2-orthonormal basis of eigenvectorss 𝜔𝑖𝑊,𝑖=1,2,,𝑚(). Any function 𝑣 in 𝑉 can thus be expanded with respect to the system 𝜔𝑖 as 𝑣=𝑚()𝑖=1𝑣,𝜔𝑖𝜔𝑖,(3.5) in particular, we have 𝑢𝑘=𝑚()𝑖=1𝑢𝑘𝑖𝜔𝑖,𝑢𝑘𝑖=𝑢𝑘,𝜔𝑖.(3.6) Moreover, let 𝑓𝑘 be the 𝐿2-orthogonal projection of 𝑓𝜃,𝑘 into 𝑊, that is, 𝑓𝑘𝑊 and 𝑓𝑘,𝑣=𝑓𝜃,𝑘,𝑣,(3.7) and set 𝑓𝑘=𝑚()𝑖=1𝑓𝑘𝑖𝜔𝑖,𝑓𝑘𝑖=𝑓𝑘,𝜔𝑖.(3.8) We are now in a position to prove the stability for 𝜃[0,1/2[

Choosing in (3.1) 𝑣=0, thus we have 1𝑢Δ𝑡𝑘𝑢𝑘1,𝑢𝜃,𝑘𝑢+𝑎𝜃,𝑘,𝑢𝜃,𝑘𝑓𝜃,𝑘,𝑢𝜃,𝑘,𝑢𝜃,𝑘𝑉.(3.9)

The inequalities (3.2) is equivalent to 1𝑢Δ𝑡𝑘𝑖𝑢𝑖𝑘1+𝜆𝑖𝜃𝑢𝑘𝑖+(1𝜃)𝑢𝑖𝑘1𝑓𝑘𝑖.(3.10)

Since 𝜔𝑖 are the eigenfunctions means 𝑎𝜔𝑖,𝜔𝑖=𝜆𝑖𝜔𝑖,𝜔𝑖=𝜆𝑖𝛿𝑖𝑖=𝜆𝑖,(3.11) for each 𝑘=0,,𝑚()1, we can rewrite (3.9) as 𝑢𝑘𝑖1(1𝜃)Δ𝑡𝜆𝑖1+𝜃Δ𝑡𝜆𝑖𝑢𝑖𝑘1+Δ𝑡1+𝜃Δ𝑡𝜆𝑖𝑓𝑘𝑖,(3.12) this inequality system stable if and only if ||||1(1𝜃)Δ𝑡𝜆𝑖1+𝜃Δ𝑡𝜆𝑖||||<1,(3.13) that is to say 22𝜃1>𝜆𝑖Δ𝑡(3.14) means 2Δ𝑡<(12𝜃)𝜆𝑖.(3.15)

So that this relation satisfied for all the eigenvalues 𝜆𝑖 of bilinear form 𝑎(,), we have to choose their highest value, and we take it for 𝜆𝑚=𝜌(𝐴) (rayon spectral)

We deduce that if 𝜃1/2 the 𝜃-scheme way is stable unconditionally (i.e., stable for all Δt). However, if 0𝜃<1/2 the 𝜃-scheme is stable unless 2Δ𝑡<(12𝜃)𝜌(𝐴).(3.16)

We can prove that there exist two positive constants 𝑐1,𝑐2 such that 𝑐12𝜆𝑚=𝑐22,(3.17) thus the method of 𝜃-scheme is stable if and only if Δ𝑡<2𝐶(12𝜃)2.(3.18)

Notice that this condition is always satisfied if 0𝜃<1/2. Hence, taking the absolute value of (3.12), we have ||𝑢𝑚𝑖||<||𝑢0𝑖||+Δ𝑡1+𝜃Δ𝑡𝜆𝑖𝑚1𝑖=1𝑓𝑘𝑖,(3.19) also we deduce that 𝑢𝑚𝑖<𝑢0𝑖+Δ𝑡1+𝜃Δ𝑡𝜆𝑖𝑚1𝑖=1𝑓𝑘𝑖.(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 𝑢𝑘22+2Δ𝑡𝑛𝑘=1𝑎𝑢𝜃,𝑘,𝑢𝜃,𝑘𝐶(𝑛)𝑛𝑘=1𝑓Δ𝑡𝜃,𝑘22.(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 such that 𝑢𝑢𝐶2||||log2,(4.1) where 𝑢 and 𝑢 are, respectively, stationery solutions to the following continue and discrete inequalities: 𝑏(𝑢,𝑣𝑢)(𝑓+𝜆𝑢,𝑣𝑢),𝑣𝐻10𝑏𝑢(Ω),(4.2),𝑣𝑢𝑓+𝜆𝑢,𝑣𝑢,𝑣𝑉,(4.3) such that 𝑏(,)=𝑎(,)+𝜆(,),(4.4)where 𝜆 is a positive constant arbitrary.We have 𝑢𝜃,𝑘=𝜃𝑢𝑘+(1𝜃)𝑢𝑘1𝜃𝑟𝜓+(1𝜃)𝑟𝜓=𝑟𝜓.Thus, we can rewrite (3.1) as, for 𝑢𝜃,𝑘𝑉𝑢𝜃,𝑘𝜃Δ𝑡,𝑣̃𝑢𝑘𝑢+𝑎𝜃,𝑘,𝑣𝑢𝜃,𝑘𝑓𝜃,𝑘+𝑢𝑘1𝜃Δ𝑡,𝑣𝑢𝜃,𝑘,𝑣𝑉.(4.5) Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities: 𝑏𝑢𝜃,𝑘,𝑣̃𝑢𝑘𝑓𝜃,𝑘+𝜇𝑢𝑘1,𝑣𝑢𝜃,𝑘,𝑣𝑉,(4.6) such that 𝑏𝑢𝜃,𝑘,𝑣𝑢𝜃,𝑘𝑢=𝜇𝜃,𝑘,𝑣𝑢𝜃,𝑘𝑢+𝑎𝜃,𝑘,𝑣𝑢𝜃,𝑘,𝑣,𝑢𝜃,𝑘𝑉,1𝜇==𝑇𝜃Δ𝑡,𝜃𝑘(4.7) where 𝑢𝜃,1 is the solution to the following discrete inequality: 𝑎𝑢𝜃,1,𝑣𝑢𝜃,1=𝑔𝑡𝑘,𝑣𝑢𝜃,1,𝑣𝑉,(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 𝑇𝐿+(Ω)𝑉𝑤𝑇(𝑤)=𝜉,(4.9) where 𝜉 is the unique solution of the following P.V.I: find 𝜉𝑉𝑏𝜉,𝑣𝜉𝑓𝜃,𝑘+𝜇𝑤,𝑣𝜉,𝑣𝑉.(4.10)

Proposition 4.2. Under the previous hypotheses and notations, if one sets 𝜃1/2, the mapping 𝑇 is a contraction in 𝐿(Ω) with rate of contraction 1/(1+𝛽𝜃Δ𝑡). Therefore, 𝑇 admits a unique fixed point which coincides with the solution of P.V. I (4.7).

Proof. For 𝑤, 𝑤 in 𝐿(Ω), we consider 𝜉=𝑇(𝑤)=𝜕(𝑓𝜃,𝑘+𝜇𝑤,𝑟𝜓) and ̃𝜉=𝑇(𝑤)=𝜕(𝑓𝜃,𝑘+𝜇𝑤,𝑟𝜓) solution to quasivariational inequalities (4.7) with right-hand side 𝐹𝜃,𝑘=𝑓𝜃,𝑘+𝜇𝑤,𝐹𝜃,𝑘=𝑓𝜃,𝑘𝑤+𝜇.
Now, setting 1𝜙=𝐹𝛽+𝜇𝜃,𝑘𝐹𝜃,𝑘,(4.11) then for 𝜉+𝜙 is solution of 𝑏𝜉𝑣+𝜙,𝜉+𝜙𝐹+𝜙𝜃,𝑘+𝑎0𝑣𝜙,𝜉+𝜙,+𝜙𝜉+𝜙𝑟𝜓+𝜙,𝑣+𝜙𝑟𝜓+𝜙,𝑣𝑉.(4.12) Also, we have 𝐹𝜃,𝑘𝐹𝜃,𝑘+𝐹𝜃,𝑘𝐹𝜃,𝑘𝐹𝜃,𝑘+𝑎0𝐹𝛽+𝜇𝜃,𝑘𝐹𝜃,𝑘𝐹𝜃,𝑘+𝑎0𝜙,(4.13) thus 𝜕𝐹𝜃,𝑘,𝑟𝜓+𝜙𝜕𝐹𝜃,𝑘+𝑎0(𝑥)𝜙,𝑟𝜓+𝜙𝜕𝐹𝜃,𝑘,𝑟𝜓+𝜙,(4.14) hence 𝜉̃𝜉+𝜙.(4.15) Similarly, interchanging the roles of 𝑤 and 𝑤, we also get ̃𝜉𝜉+𝜙.(4.16) Finally, this yields 𝜕𝐹𝜃,𝑘,𝑟𝜓𝜕𝐹𝜃,𝑘,𝑟𝜓1𝐹𝛽+𝜇𝜃,𝑘𝐹𝜃,𝑘1𝑓𝛽+𝜇𝜃,𝑘+𝜇𝑤𝑓𝜃,𝑘𝑤𝜇𝑤𝑤1𝑤1+𝛽𝜃Δ𝑡𝑤,(4.17) which completes the proof.

Remark 4.3. If we set 0𝜃<1/2, the mapping 𝑇 is a contraction in 𝐿(Ω) with rate of contraction 2/(2+𝛽𝜃(12𝜃)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/(12𝜃))2.
Thus it can be easily show that 𝜕𝐹𝜃,𝑘,𝑟𝜓𝜕𝐹𝜃,𝑘,𝑟𝜓1𝑤1+𝛽𝜃Δ𝑡𝑤2𝑤2+𝛽𝜃(12𝜃)𝜌(𝐴)𝑤11+𝛽𝜃(12𝜃)/2𝐶2𝑤𝑤=2𝐶22𝐶2𝑤+𝛽𝜃(12𝜃)𝑤,(4.18) also it can be found that 𝜕𝐹𝜃,𝑘,𝑟𝜓𝜕𝐹𝜃,𝑘,𝑟𝜓1(1+𝛽𝜃12𝜃)/2𝐶2𝑤𝑤=2𝐶22𝐶2𝑤+𝛽𝜃(12𝜃)𝑤,(4.19) thus the mapping 𝑇 is a contraction in 𝐿(Ω) with rate of contraction (2𝐶2)/(2𝐶2+𝛽𝜃(12𝜃)). Therefore, 𝑇 admits a unique fixed point which coincides with the solution of P.V.I (4.7) 𝑇(𝑤)T𝑤1𝐹𝛽+𝜇𝜃,𝑘𝐹𝜃,𝑘=1𝑓𝛽+𝜇𝜃,𝑘+𝜇𝑤𝑓𝜃,𝑘𝑤𝜇𝜇𝑤𝛽+𝜇𝑤1𝑤1+𝛽𝜃Δ𝑡𝑤.(4.20) This completes the proof.

4.2. Discrete Algorithm

Starting from 𝑢0=𝑢0 (initial data) and the 𝑢𝜃,1 solution of problem (4.7), we introduce the following discrete algorithm: 𝑢𝜃,𝑘=𝑇𝑢𝑘1,𝑘=1,,𝑛,(4.21) where 𝑢𝜃,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 𝜃1/2𝑢𝜃,𝑘𝑢11+𝛽𝜃Δ𝑡𝑘𝑢𝑢0,(4.22) and one has for 𝑢𝜃,𝑘𝑢2𝐶22𝐶2+𝛽𝜃(12𝜃)𝑘𝑢𝑢01for0𝜃<2.(4.23)

Proof. we set a first case 𝜃1/2, and we have 𝑢=𝑇𝑢,𝑢𝜃,1𝑢=𝑇𝑢0𝑇𝑢1𝑢1+𝛽𝜃Δ𝑡0𝑢.(4.24) for 𝑘2, we use the Bensoussan-Lions' algorithm (𝑢𝑘=𝑇𝑢𝑘1,𝑘=1,,𝑛) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that 𝑢𝜃,𝑘𝑢11+𝛽𝜃Δ𝑡𝑘𝑢0𝑢,(4.25) so 𝑢𝜃,𝑘+1𝑢=𝑇𝑢𝑘𝑇𝑢1𝑢1+𝛽𝜃Δ𝑡𝑘𝑢,(4.26) thus 𝑢𝜃,𝑘+1𝑢11+𝛽𝜃Δ𝑡𝑘+1𝑢0𝑢,(4.27) for a second case 0𝜃<1/2, it can be easily shown that 𝑢𝜃,𝑘𝑢2𝐶22𝐶2+𝛽𝜃(12𝜃)𝑘𝑢0𝑢.(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 𝐿-norm for parabolic variational inequalities

Now, we evaluate the variation in 𝐿 between 𝑢𝜃(𝑇,𝑥), the discrete solution calculated at the moment 𝑇=𝑛Δ𝑡 and 𝑢, 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 𝜃1/2, 𝑢𝜃,𝑛𝑢𝐶2||||log2+11+𝛽𝜃Δ𝑡𝑛,(4.29) and for the second case 0𝜃<1/2, 𝑢𝜃,𝑛𝑢𝐶2||||log2+2𝐶22𝐶2+𝛽𝜃(12𝜃)𝑛,(4.30) where 𝐶 is a constant independent of and 𝑘.

Proof. We have 𝑢𝜃,𝑘(𝑥)=𝑢][(𝑡,𝑥)for𝑡(𝑘1)Δ𝑡;𝑘Δ𝑡,(4.31) thus 𝑢𝜃,𝑛(𝑥)=𝑢(𝑇,𝑥),(4.32) then 𝑢𝜃(𝑇,𝑥)𝑢𝐿(Ω)=𝑢𝜃,𝑛𝑢𝐿(Ω)𝑢𝜃,𝑛𝑢𝐿(Ω)+𝑢𝑢𝐿(Ω).(4.33) Using, Theorem 4.1 and Proposition 4.5, we have for 𝜃1/2, 𝑢𝜃,𝑛𝑢𝐶2||||log2+11+𝛽𝜃Δ𝑡𝑛,(4.34) and for 0𝜃<1/2, we have 𝑢𝜃,𝑛𝑢𝐶2||||log2+2𝐶22𝐶2+𝛽𝜃(12𝜃)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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