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)∈𝐿20,𝑇,𝐻10(Ω),𝑣(𝑡,𝑥)≤𝜓(𝑡,𝑥),𝑣(0,𝑥)=𝑣0inΩ,(1.1) with 𝜓∈𝐿20,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=−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),𝑥∈Ω,1≤𝑖,𝑗≤𝑁 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;𝜉∈ℝ𝑁,𝛾>0,𝑥∈Ω,(1.6)𝑓 is a regular functions satisfying 𝑓∈𝐿2(0,𝑇,𝐿∞(Ω))∩𝐶10,𝑇,𝐻−1(Ω),𝑓≥0.(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 𝑉ℎ⊂𝐻10. In the second step, we discretize the problem with respect to time using the 𝜃-scheme. Therefore, we search a sequence of elements 𝑢𝑛ℎ∈𝑉ℎ 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 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,𝑣−𝑢+𝑎(𝑢,𝑣−𝑢)≥(𝑓,𝑣−𝑢),𝑣∈𝐾,(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 𝑢∈𝐾(𝑢). Moreover, one has 𝑢∈𝐿20,𝑇;𝐻10,(Ω)𝜕𝑢𝜕𝑡∈𝐿20,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)∩𝐶2Ω,∇=𝑁𝑖=1𝜕𝜕𝑥𝑖.(2.4)

2.1. The Discrete Problem

Let us assume that Ω can be decomposed into triangles and 𝜏ℎ denotes the set of all the elements ℎ>0, where ℎ is the mesh size. We assume that the family 𝜏ℎ 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 𝑉ℎ of finite element: 𝑉ℎ=𝑣ℎ∈𝐿20,𝑇,𝐻10(Ω)∩𝐶0,𝑇,𝐻10Ω,suchthat𝑣ℎ|𝑘∈𝑃1,𝑘∈𝜏ℎ,𝑣ℎ≤𝑟ℎ𝜓,𝑣ℎ(⋅,0)=𝑣ℎ0.inΩ.(2.5)

We consider 𝑟ℎ to be the usual interpolation operator defined by 𝑣∈𝐿20,𝑇,𝐻10(Ω)∩𝐶0,𝑇,𝐻10Ω,𝑟ℎ𝑣=𝑚(ℎ)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 ℎ.

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 𝑉ℎ⊂𝐻10(Ω). Then, there exists two constants 𝛼>0 and 𝜆>0 such that 𝑎𝑢ℎ,𝑢ℎ‖‖𝑢+𝜆ℎ‖‖2‖‖𝑢≥𝛼ℎ‖‖1,(2.7) where ‖‖𝑏𝜆=𝑗‖‖2∞+𝛾2𝛾2+‖‖𝑎0‖‖∞𝛾,𝛼=2.(2.8)

Proof. The bilinear form 𝑎(⋅,⋅) is defined by 𝑎𝑢ℎ,𝑢ℎ=Ω𝑁𝑖𝑗=1𝑎𝑖𝑗(𝑥)𝜕𝑢ℎ𝜕𝑥𝑖𝜕𝑢ℎ𝜕𝑥𝑗+𝑁𝑗=1𝑏𝑗(𝑥)𝜕𝑢ℎ𝜕𝑥𝑗𝑢ℎ+𝑎0(𝑥)𝑢2ℎ𝑑𝑥,(2.9) under assumption (1.6), we have 𝑁𝑖𝑗=1Ω𝑎𝑖𝑗(𝑥)𝜕𝑢ℎ𝜕𝑥𝑖𝜕𝑢ℎ𝜕𝑥𝑗>𝛾𝑁𝑖=1Ω𝜕𝑢ℎ𝜕𝑥𝑖2‖‖=𝛾∇𝑢ℎ‖‖22,(2.10) and since |||||𝑁𝑗=1Ω𝑏𝑗(𝑥)𝜕𝑢ℎ𝜕𝑥𝑗𝑢ℎ|||||≤sup𝑗||𝑏𝑗||𝑁𝑗=1Ω||||𝜕𝑢ℎ𝜕𝑥𝑗𝑢ℎ||||≤‖‖𝑏𝑗‖‖∞‖‖∇𝑢ℎ‖‖2‖‖𝑢ℎ‖‖2,(2.11) then we make use of the algebraic inequality 1𝑎𝑏≤2𝑎2+𝑏2,∀𝑎,𝑏∈ℝ,∀𝛾>0,(2.12) and choosing ‖‖𝑎=∇𝑢ℎ‖‖2⋅√‖‖𝑏𝛾,𝑏=𝑗‖‖∞√𝛾‖‖𝑢ℎ‖‖2,(2.13) then we end up with |||||𝑁𝑗=1Ω𝑏𝑗(𝑥)𝜕𝑢ℎ𝜕𝑥𝑗𝑢ℎ|||||𝛾≥−2‖‖∇𝑢ℎ‖‖22+‖‖𝑏𝑗‖‖∞⋅‖‖𝑢ℎ‖‖22𝛾,(2.14) so we get 𝑎𝑢ℎ,𝑢ℎ‖‖≥𝛾∇𝑢ℎ‖‖22−𝛾2‖‖∇𝑢ℎ‖‖22+‖‖𝑏𝑗‖‖∞‖‖𝑢2𝛾ℎ‖‖2−‖‖𝑎0‖‖∞‖‖𝑢ℎ‖‖22.(2.15) It can easily verified that 𝑎𝑢ℎ,𝑢ℎ≥𝛾2‖‖∇𝑢ℎ‖‖2+‖‖𝑢ℎ‖‖22−‖‖𝑎𝑗‖‖2∞+𝛾2𝛾2+‖‖𝑎0‖‖∞‖‖𝑢ℎ‖‖22.(2.16) Consequently, we deduce from above that 𝑎𝑢ℎ,𝑢ℎ‖‖𝑢+𝜆ℎ‖‖22‖‖𝑢≥𝛼ℎ‖‖21𝛾suchthat𝛼=2‖‖𝑎,𝜆=𝑗‖‖2∞+𝛾2𝛾2+‖‖𝑎0‖‖∞.(2.17)

We can identify the following result on the time energy behavior: 𝐸ℎ(𝑡)=Ω𝑢2ℎ𝑑𝑥.(2.18)

Setting 𝑣=0 on (2.1) and after discretization by the finite element in the 𝑉ℎ, we have the semidiscretization problem 𝜕𝑢ℎ𝜕𝑡,𝑢ℎ𝑢+𝑎ℎ,𝑢ℎ=12Ω𝜕𝑢2ℎ𝑢𝜕𝑡𝑑𝑥+𝑎ℎ,𝑢ℎ≤𝑓,𝑢ℎ.(2.19)

Using Theorem 2.3, we deduce that12Ω𝑑𝑢2ℎ𝑢𝑑𝑡𝑑𝑥+𝑎ℎ,𝑢ℎ≥12𝑑𝑑𝑡Ω𝑢2ℎ‖‖𝑢𝑑𝑥+𝛼ℎ‖‖21‖‖𝑢−𝜆ℎ‖‖22=12𝑑𝑑𝑡Ω𝑢2ℎ‖‖𝑢𝑑𝑥+𝛼ℎ‖‖22‖‖+𝛼∇𝑢ℎ‖‖22‖‖𝑢−𝜆ℎ‖‖22=12𝑑𝐸𝑑𝑡ℎ(𝑡)+2(𝛼−𝜆)𝐸ℎ‖‖(𝑡)+2𝛼∇𝑢ℎ‖‖22≥1𝑑𝑥2𝑑𝐸𝑑𝑡ℎ(𝑡)+2(𝛼−𝜆)𝐸ℎ(𝑡)+2𝛼𝐶2∗Ω𝑢2ℎ.𝑑𝑥(2.20)

Thus, we have𝑑𝑑𝑡Ω𝑢2ℎ‖‖𝑢𝑑𝑥+𝛼ℎ‖‖21‖‖𝑢−𝜆ℎ‖‖22≥𝑑𝐸𝑑𝑡ℎ𝛼(𝑡)+2𝛼−𝜆+𝐶2∗𝐸ℎ(𝑡).(2.21)

Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find𝑓,𝑢ℎ=Ω𝑓(𝑥,𝑡)𝑢ℎ(𝑥,𝑡)𝑑𝑥≤‖𝑓‖2‖‖𝑢ℎ‖‖2.(2.22)

So that 𝑑𝐸𝑑𝑡ℎ𝛼(𝑡)+2𝛼−𝜆+𝐶2∗𝐸ℎ(𝑡)≤2‖𝑓‖2‖‖𝑢ℎ‖‖2.(2.23)

Using Young’s inequality𝑎𝑏≤𝜀𝑎2+1𝑏4𝜀2,∀𝑎,𝑏∈ℝ,∀𝜀>0.(2.24)

Thus, we obtain𝑑𝐸𝑑𝑡ℎ𝛼(𝑡)+2𝛼−𝜆+𝐶2∗𝐸ℎ(𝑡)≤2𝜀𝐸ℎ1(𝑡)+2𝜀‖𝑓‖22,(2.25)

taking 𝜂=𝛼−𝜆+𝛼/𝐶2∗, thus we have 𝑑𝐸𝑑𝑡ℎ(𝑡)+2(𝜂−𝜀)𝐸ℎ1(𝑡)≤2𝜀‖𝑓‖22.(2.26)

Or, equivalently 𝑒2(𝜂−𝜀)𝑡𝐸ℎ(𝑡)≤1𝑒2𝜀2(𝜂−𝜀)𝑡Ω(𝑓(𝑥,𝑡))2𝑑𝑥.(2.27)

Integrating the last inequality from 0 to 𝑡, we get 𝐸ℎ(𝑡)≤𝑒−2(𝜂−𝜀)𝑡𝐸ℎ(10)+2𝜀𝑡0𝑒2(𝜂−𝜀)(𝑠−𝑡)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 𝜕𝑢ℎ𝜕𝑡,𝑣ℎ−𝑢ℎ𝑢+𝑎ℎ,𝑣ℎ−𝑢ℎ≥𝑓,𝑣ℎ−𝑢ℎ,𝑣ℎ∈𝑉ℎ.(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 𝑢𝑘ℎ−𝑢ℎ𝑘−1,𝑣ℎ−𝑢ℎ𝜃,𝑘𝑢+Δ𝑡⋅𝑎ℎ𝜃,𝑘,𝑣ℎ−𝑢ℎ𝜃,𝑘𝑓≥Δ𝑡⋅𝜃,𝑘,𝑣ℎ−𝑢ℎ𝜃,𝑘,𝑣ℎ∈𝑉ℎ,(3.2) where 𝑢ℎ𝜃,𝑘=𝜃𝑢𝑘ℎ+(1−𝜃)𝑢ℎ𝑘−1,𝑓𝜃,𝑘𝑡=𝜃𝑓𝑘𝑡+(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 𝑎(⋅,⋅). 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 𝑎(⋅,⋅) satisfying 𝜔𝑗∈𝐿2,𝜔𝑗𝜔≠0:𝑎𝑗,𝑣ℎ=𝜆𝑗𝜔𝑗,𝑣ℎ,∀𝑣ℎ∈𝑉ℎ.(3.4) The corresponding eigenfunctions {𝜔𝑗} form a complete orthonormal basis in 𝐿2(Ω). In analogous way, when considering the finite dimensional problem in 𝑊ℎ, we find a sequence of eigenvalues 𝛿≤𝜆1ℎ≤𝜆2ℎ≤⋯≤𝜆𝑚(ℎ) and 𝐿2-orthonormal basis of eigenvectorss 𝜔𝑖ℎ∈𝑊ℎ,𝑖=1,2,…,𝑚(ℎ). Any function 𝑣ℎ in 𝑉ℎ can thus be expanded with respect to the system 𝜔𝑖ℎ as 𝑣ℎ=𝑚(ℎ)𝑖=1𝑣ℎ,𝜔𝑖ℎ𝜔𝑖ℎ,(3.5) in particular, we have 𝑢𝑘ℎ=𝑚(ℎ)𝑖=1𝑢𝑘𝑖𝜔𝑖ℎ,𝑢𝑘𝑖=𝑢𝑘ℎ,𝜔𝑖ℎ.(3.6) Moreover, let 𝑓𝑘ℎ be the 𝐿2-orthogonal projection of 𝑓𝜃,𝑘 into 𝑊ℎ, that is, 𝑓𝑘ℎ∈𝑊ℎ and 𝑓𝑘ℎ,𝑣ℎ=𝑓𝜃,𝑘,𝑣ℎ,(3.7) and set 𝑓𝑘ℎ=𝑚(ℎ)𝑖=1𝑓𝑘𝑖𝜔𝑖ℎ,𝑓𝑘𝑖=𝑓𝑘ℎ,𝜔𝑖ℎ.(3.8) We are now in a position to prove the stability for 𝜃∈[0,1/2[

Choosing in (3.1) 𝑣ℎ=0, thus we have 1𝑢Δ𝑡𝑘ℎ−𝑢ℎ𝑘−1,𝑢ℎ𝜃,𝑘𝑢+𝑎ℎ𝜃,𝑘,𝑢ℎ𝜃,𝑘≤𝑓𝜃,𝑘,𝑢ℎ𝜃,𝑘,𝑢ℎ𝜃,𝑘∈𝑉ℎ.(3.9)

The inequalities (3.2) is equivalent to 1𝑢Δ𝑡𝑘𝑖−𝑢𝑖𝑘−1+𝜆𝑖ℎ𝜃𝑢𝑘𝑖+(1−𝜃)𝑢𝑖𝑘−1≤𝑓𝑘𝑖.(3.10)

Since 𝜔𝑖ℎ are the eigenfunctions means 𝑎𝜔𝑖ℎ,𝜔𝑖ℎ=𝜆𝑖ℎ𝜔𝑖ℎ,𝜔𝑖ℎ=𝜆𝑖ℎ⋅𝛿𝑖𝑖=𝜆𝑖ℎ,(3.11) for each 𝑘=0,…,𝑚(ℎ)−1, we can rewrite (3.9) as 𝑢𝑘𝑖≤1−(1−𝜃)⋅Δ𝑡⋅𝜆𝑖ℎ1+𝜃Δ𝑡⋅𝜆𝑖ℎ𝑢𝑖𝑘−1+Δ𝑡1+𝜃Δ𝑡⋅𝜆𝑖ℎ𝑓𝑘𝑖,(3.12) this inequality system stable if and only if ||||1−(1−𝜃)⋅Δ𝑡⋅𝜆𝑖ℎ1+𝜃Δ𝑡⋅𝜆𝑖ℎ||||<1,(3.13) that is to say 22𝜃−1>−𝜆𝑖ℎ⋅Δ𝑡(3.14) means 2Δ𝑡<(1−2𝜃)𝜆𝑖ℎ.(3.15)

So that this relation satisfied for all the eigenvalues 𝜆𝑖ℎ of bilinear form 𝑎(⋅,⋅), we have to choose their highest value, and we take it for 𝜆𝑚ℎ=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 𝑐1ℎ2≤𝜆𝑚ℎ=𝑐2ℎ−2,(3.17) thus the method of 𝜃-scheme is stable if and only if Δ𝑡<2𝐶ℎ(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+𝜃Δ𝑡⋅𝜆𝑖ℎ𝑚−1𝑖=1𝑓𝑘𝑖,(3.19) also we deduce that ‖‖𝑢𝑚𝑖‖‖∞<‖‖𝑢0𝑖‖‖∞+‖‖‖Δ𝑡1+𝜃Δ𝑡⋅𝜆𝑖ℎ‖‖‖∞𝑚−1𝑖=1‖‖𝑓𝑘𝑖‖‖∞.(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 ‖‖𝑢𝑘ℎ‖‖22+2Δ𝑡𝑛𝑘=1𝑎𝑢ℎ𝜃,𝑘,𝑢ℎ𝜃,𝑘≤𝐶(𝑛)𝑛𝑘=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 ‖‖𝑢∞−𝑢∞ℎ‖‖∞≤𝐶ℎ2||||logℎ2,(4.1) where 𝑢∞ and 𝑢∞ℎ are, respectively, stationery solutions to the following continue and discrete inequalities: 𝑏(𝑢∞,𝑣−𝑢∞)≥(𝑓+𝜆𝑢∞,𝑣−𝑢∞),𝑣∈𝐻10𝑏𝑢(Ω),(4.2)∞ℎ,𝑣ℎ−𝑢∞ℎ≥𝑓+𝜆𝑢∞,𝑣ℎ−𝑢∞ℎ,𝑣ℎ∈𝑉ℎ,(4.3) such that 𝑏(⋅,⋅)=𝑎(⋅,⋅)+𝜆(⋅,⋅),(4.4)where 𝜆 is a positive constant arbitrary.We have 𝑢ℎ𝜃,𝑘=𝜃𝑢𝑘ℎ+(1−𝜃)𝑢ℎ𝑘−1≤𝜃𝑟ℎ𝜓+(1−𝜃)𝑟ℎ𝜓=𝑟ℎ𝜓.Thus, we can rewrite (3.1) as, for 𝑢ℎ𝜃,𝑘∈𝑉ℎ𝑢ℎ𝜃,𝑘𝜃Δ𝑡,𝑣ℎ−̃𝑢𝑘ℎ𝑢+𝑎ℎ𝜃,𝑘,𝑣ℎ−𝑢ℎ𝜃,𝑘≥𝑓𝜃,𝑘+𝑢ℎ𝑘−1𝜃Δ𝑡,𝑣ℎ−𝑢ℎ𝜃,𝑘,𝑣ℎ∈𝑉ℎ.(4.5) Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities: 𝑏𝑢ℎ𝜃,𝑘,𝑣ℎ−̃𝑢𝑘ℎ≥𝑓𝜃,𝑘+𝜇𝑢ℎ𝑘−1,𝑣ℎ−𝑢ℎ𝜃,𝑘,𝑣ℎ∈𝑉ℎ,(4.6) such that 𝑏𝑢ℎ𝜃,𝑘,𝑣ℎ−𝑢ℎ𝜃,𝑘𝑢=𝜇ℎ𝜃,𝑘,𝑣ℎ−𝑢ℎ𝜃,𝑘𝑢+𝑎ℎ𝜃,𝑘,𝑣ℎ−𝑢ℎ𝜃,𝑘,𝑣ℎ,𝑢ℎ𝜃,𝑘∈𝑉ℎ,1𝜇==𝑇𝜃Δ𝑡,𝜃𝑘(4.7) where 𝑢ℎ𝜃,1 is the solution to the following discrete inequality: 𝑎𝑢ℎ𝜃,1,𝑣ℎ−𝑢ℎ𝜃,1=𝑔𝑡𝑘,𝑣ℎ−𝑢ℎ𝜃,1,𝑣ℎ∈𝑉ℎ,(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 𝑇ℎ∶𝐿∞+(Ω)⟶𝑉ℎ𝑤⟶𝑇ℎ(𝑤)=𝜉ℎ,(4.9) where 𝜉ℎ is the unique solution of the following P.V.I: find 𝜉ℎ∈𝑉ℎ𝑏𝜉ℎ,𝑣ℎ−𝜉ℎ≥𝑓𝜃,𝑘+𝜇𝑤,𝑣ℎ−𝜉ℎ,𝑣ℎ∈𝑉ℎ.(4.10)

Proposition 4.2. Under the previous hypotheses and notations, if one sets 𝜃≥1/2, the mapping 𝑇ℎ is a contraction in 𝐿∞(Ω) with rate of contraction 1/(1+𝛽⋅𝜃⋅Δ𝑡). Therefore, 𝑇ℎ 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 𝜉ℎ=𝑇ℎ(𝑤)=𝜕(𝑓𝜃,𝑘+𝜇𝑤,𝑟ℎ𝜓) and ̃𝜉ℎ=𝑇ℎ(𝑤)=𝜕(𝑓𝜃,𝑘+𝜇𝑤,𝑟ℎ𝜓) solution to quasivariational inequalities (4.7) with right-hand side 𝐹𝜃,𝑘=𝑓𝜃,𝑘+𝜇𝑤ℎ,𝐹𝜃,𝑘=𝑓𝜃,𝑘𝑤+𝜇ℎ.
Now, setting 1𝜙=‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,𝑘‖‖∞,(4.11) then for 𝜉ℎ+𝜙 is solution of 𝑏𝜉ℎ𝑣+𝜙,ℎ−𝜉+𝜙ℎ≥𝐹+𝜙𝜃,𝑘+𝑎0𝑣𝜙,ℎ−𝜉+𝜙ℎ,+𝜙𝜉+𝜙≤𝑟ℎ𝜓+𝜙,𝑣ℎ+𝜙≤𝑟ℎ𝜓+𝜙,∀𝑣ℎ∈𝑉ℎ.(4.12) Also, we have 𝐹𝜃,𝑘≤𝐹𝜃,𝑘+‖‖𝐹𝜃,𝑘−𝐹𝜃,𝑘‖‖∞≤𝐹𝜃,𝑘+𝑎0‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,𝑘‖‖∞≤𝐹𝜃,𝑘+𝑎0𝜙,(4.13) thus 𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ𝜓+𝜙≤𝜕ℎ𝐹𝜃,𝑘+𝑎0(𝑥)𝜙,𝑟ℎ𝜓+𝜙≤𝜕ℎ𝐹𝜃,𝑘,𝑟𝜓+𝜙,(4.14) hence 𝜉ℎ≤̃𝜉ℎ+𝜙.(4.15) Similarly, interchanging the roles of 𝑤 and 𝑤, we also get ̃𝜉ℎ≤𝜉ℎ+𝜙.(4.16) Finally, this yields ‖‖𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ𝜓−𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ‖‖𝜓∞≤1‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,𝑘‖‖∞≤1‖‖𝑓𝛽+𝜇𝜃,𝑘+𝜇𝑤−𝑓𝜃,𝑘𝑤‖‖−𝜇∞‖‖𝑤‖‖𝑤−∞≤1‖‖𝑤‖‖1+𝛽𝜃Δ𝑡𝑤−∞,(4.17) which completes the proof.

Remark 4.3. If we set 0≤𝜃<1/2, the mapping 𝑇ℎ is a contraction in 𝐿∞(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 ‖‖𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ𝜓−𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ‖‖𝜓∞≤1‖‖𝑤‖‖1+𝛽𝜃Δ𝑡𝑤−∞≤2‖‖𝑤‖‖2+𝛽𝜃(1−2𝜃)𝜌(𝐴)𝑤−∞≤11+𝛽𝜃(1−2𝜃)/2𝐶ℎ2‖‖𝑤‖‖𝑤−∞=2𝐶ℎ22𝐶ℎ2‖‖𝑤‖‖+𝛽𝜃(1−2𝜃)𝑤−∞,(4.18) also it can be found that ‖‖𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ𝜓−𝜕ℎ𝐹𝜃,𝑘,𝑟ℎ‖‖𝜓∞≤1(1+𝛽𝜃1−2𝜃)/2𝐶ℎ2‖‖𝑤‖‖𝑤−∞=2𝐶ℎ22𝐶ℎ2‖‖𝑤‖‖+𝛽𝜃(1−2𝜃)𝑤−∞,(4.19) thus the mapping 𝑇ℎ is a contraction in 𝐿∞(Ω) with rate of contraction (2𝐶ℎ2)/(2𝐶ℎ2+𝛽𝜃(1−2𝜃)). Therefore, 𝑇ℎ admits a unique fixed point which coincides with the solution of P.V.I (4.7) ‖‖𝑇ℎ(𝑤)−Tℎ𝑤‖‖∞≤1‖‖𝐹𝛽+𝜇𝜃,𝑘−𝐹𝜃,𝑘‖‖∞=1‖‖𝑓𝛽+𝜇𝜃,𝑘+𝜇𝑤−𝑓𝜃,𝑘𝑤‖‖−𝜇∞≤𝜇‖‖𝑤‖‖𝛽+𝜇𝑤−∞≤1‖‖𝑤‖‖1+𝛽𝜃Δ𝑡𝑤−∞.(4.20) This completes the proof.

4.2. Discrete Algorithm

Starting from 𝑢0ℎ=𝑢0ℎ (initial data) and the 𝑢ℎ𝜃,1 solution of problem (4.7), we introduce the following discrete algorithm: 𝑢ℎ𝜃,𝑘=𝑇ℎ𝑢ℎ𝑘−1,𝑘=1,…,𝑛,(4.21) where 𝑢ℎ𝜃,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≥1/2‖‖𝑢ℎ𝜃,𝑘−𝑢∞ℎ‖‖∞≤11+𝛽𝜃Δ𝑡𝑘‖‖𝑢∞ℎ−𝑢ℎ0‖‖∞,(4.22) and one has for ‖‖𝑢ℎ𝜃,𝑘−𝑢∞ℎ‖‖∞≤2𝐶ℎ22𝐶ℎ2+𝛽𝜃(1−2𝜃)𝑘‖‖𝑢∞ℎ−𝑢ℎ0‖‖∞1for0≤𝜃<2.(4.23)

Proof. we set a first case 𝜃≥1/2, and we have 𝑢∞ℎ=𝑇ℎ𝑢∞ℎ,‖‖𝑢ℎ𝜃,1−𝑢∞ℎ‖‖∞=‖‖𝑇ℎ𝑢0ℎ−𝑇ℎ𝑢∞ℎ‖‖∞≤1‖‖𝑢1+𝛽𝜃Δ𝑡0ℎ−𝑢∞ℎ‖‖∞.(4.24) for 𝑘≥2, we use the Bensoussan-Lions' algorithm (𝑢𝑘ℎ=𝑇ℎ𝑢ℎ𝑘−1,𝑘=1,…,𝑛) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that ‖‖𝑢ℎ𝜃,𝑘−𝑢∞ℎ‖‖∞≤11+𝛽𝜃Δ𝑡𝑘‖‖𝑢0ℎ−𝑢∞ℎ‖‖∞,(4.25) so ‖‖𝑢ℎ𝜃,𝑘+1−𝑢∞ℎ‖‖∞=‖‖𝑇ℎ𝑢𝑘ℎ−𝑇ℎ𝑢∞ℎ‖‖∞≤1‖‖𝑢1+𝛽𝜃Δ𝑡𝑘ℎ−𝑢∞ℎ‖‖∞,(4.26) thus ‖‖𝑢ℎ𝜃,𝑘+1−𝑢∞ℎ‖‖∞≤11+𝛽𝜃Δ𝑡𝑘+1‖‖𝑢0ℎ−𝑢∞ℎ‖‖∞,(4.27) for a second case 0≤𝜃<1/2, it can be easily shown that ‖‖𝑢ℎ𝜃,𝑘−𝑢∞ℎ‖‖∞≤2𝐶ℎ22𝐶ℎ2+𝛽𝜃(1−2𝜃)𝑘‖‖𝑢0ℎ−𝑢∞ℎ‖‖∞.(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 𝑢𝜃ℎ(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 𝜃≥1/2, ‖‖𝑢ℎ𝜃,𝑛−𝑢∞‖‖∞ℎ≤𝐶2||||logℎ2+11+𝛽𝜃Δ𝑡𝑛,(4.29) and for the second case 0≤𝜃<1/2, ‖‖𝑢ℎ𝜃,𝑛−𝑢∞‖‖∞ℎ≤𝐶2||||logℎ2+2𝐶ℎ22𝐶ℎ2+𝛽𝜃(1−2𝜃)𝑛,(4.30) where 𝐶 is a constant independent of ℎ and 𝑘.

Proof. We have 𝑢ℎ𝜃,𝑘(𝑥)=𝑢ℎ][(𝑡,𝑥)for𝑡∈(𝑘−1)Δ𝑡;𝑘Δ𝑡,(4.31) thus 𝑢ℎ𝜃,𝑛(𝑥)=𝑢ℎ(𝑇,𝑥),(4.32) then ‖‖𝑢𝜃ℎ(𝑇,𝑥)−𝑢∞‖‖𝐿∞(Ω)=‖‖𝑢ℎ𝜃,𝑛−𝑢∞‖‖𝐿∞(Ω)≤‖‖𝑢ℎ𝜃,𝑛−𝑢∞ℎ‖‖𝐿∞(Ω)+‖‖𝑢∞ℎ−𝑢∞‖‖𝐿∞(Ω).(4.33) Using, Theorem 4.1 and Proposition 4.5, we have for 𝜃≥1/2, ‖‖𝑢ℎ𝜃,𝑛−𝑢∞‖‖∞ℎ≤𝐶2||||logℎ2+11+𝛽𝜃Δ𝑡𝑛,(4.34) and for 0≤𝜃<1/2, we have ‖‖𝑢ℎ𝜃,𝑛−𝑢∞‖‖∞ℎ≤𝐶2||||logℎ2+2𝐶ℎ22𝐶ℎ2+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.