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

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)=𝑣₀inΩ,(1.1) with 𝜓∈𝐿²0,𝑇,𝑊^2,∞(Ω).(1.2)

We consider the following problem, find 𝑢∈𝐾 solution of 𝜕𝑢𝜕𝑢𝑡(+𝑡,𝑥𝐴)𝑢=≤0𝑓ininΓΣ,,(1.3) where Σ is a set in ℝ×ℝ^𝑁 defined as Σ=Ω×[0,𝑇] with 𝑇^⋅<+∞, and Ω is convex domain in ℝ^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=−^𝑁_𝑖_𝑗=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥_𝑗+^𝑁_𝑗=1𝑏_𝑗(𝑥)𝜕𝑢𝜕𝑥_𝑗+𝑎₀(𝑥)𝑢,(1.4) and whose coefficients: 𝑎_𝑖,𝑗(𝑥),𝑏_𝑗(𝑥),𝑎₀(𝑥)∈𝐿^∞(Ω)∩𝐶₂(Ω),𝑥∈Ω,1≤𝑖,𝑗≤𝑁 are sufficiently smooth functions and satisfy the following conditions: 𝑎_𝑖𝑗(𝑥)=𝑎_𝑗𝑖(𝑥);𝑎₀(𝑥)≥𝛽>0,𝛽isaconstant,(1.5)_𝑁𝑖_𝑗=1𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≥𝛾||𝜉||²;𝜉∈ℝ^𝑁,𝛾>0,𝑥∈Ω,(1.6)𝑓 is a regular functions satisfying 𝑓∈𝐿²(0,𝑇,𝐿^∞(Ω))∩𝐶¹0,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 𝑉^ℎ⊂𝐻¹⁰. In the second step, we discretize the problem with respect to time using the 𝜃-scheme. Therefore, we search a sequence of elements 𝑢^𝑛ℎ∈𝑉^ℎ which approaches 𝑢^𝑛(𝑡_𝑛),𝑡_𝑛=𝑛Δ𝑡, with initial data 𝑢^0ℎ=𝑢_0ℎ. 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: 𝜕𝜕𝑢𝑡,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)=_Ω^𝑁_𝑖𝑗=1𝑎_𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥_𝑖𝜕𝑣𝜕𝑥_𝑗+^𝑁_𝑗=1𝑏_𝑗(𝑥)𝜕𝑢𝜕𝑥_𝑗𝑣+𝑎₀(𝑥)𝑢𝑣𝑑𝑥,(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 ℝ^𝑁, 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𝜕𝜕𝑥_𝑖.(2.4)

2.1. The Discrete Problem

Let us assume that Ω can be decomposed into triangles and 𝜏_ℎ denotes the set of all the elements ℎ>0, where ℎ is the mesh size. We assume that the family 𝜏_ℎ is regular and quasi-uniform, and we consider the usual basis of affine functions 𝜑_𝑖, 𝑖={1,…,𝑚(ℎ)} defined by 𝜑_𝑖(𝑀_𝑗)=𝛿_𝑖𝑗, where 𝑀_𝑗 is a vertex of the considered triangulation. We introduce the following discrete spaces 𝑉^ℎ of finite element: 𝑉_ℎ=𝑣𝑣_ℎℎ∈𝐿²0,𝑇,𝐻¹⁰(Ω)∩𝐶0,𝑇,𝐻¹⁰Ω,suchthat𝑣_ℎ|_𝑘∈𝑃₁,𝑘∈𝜏_ℎ,≤𝑟_ℎ𝜓,𝑣_ℎ(⋅,0)=𝑣_ℎ0inΩ..(2.5)

We consider 𝑟_ℎ to be the usual interpolation operator defined by 𝑣∈𝐿²0,𝑇,𝐻¹⁰(Ω)∩𝐶0,𝑇,𝐻¹⁰Ω,𝑟_ℎ𝑣=^𝑚(ℎ)_𝑖=1𝑣𝑀_𝑖𝜑_𝑖(𝑥).(2.6)

The Discrete Maximum Principle Assumption (see [10])
The matrix whose coefficients 𝑎(𝜑_𝑖,𝜑_𝑗) 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 𝑉^ℎ⊂𝐻¹⁰(Ω). Then, there exists two constants 𝛼>0 and 𝜆>0 such that 𝑎𝑢_ℎ,𝑢_ℎ+𝜆‖‖𝑢_ℎ‖‖₂≥𝛼‖‖𝑢_ℎ‖‖₁,(2.7) where 𝜆=‖‖𝑏^𝑗‖‖^2∞2𝛾+𝛾2+‖‖𝑎⁰‖‖_∞,𝛼=𝛾2.(2.8)

Proof. The bilinear form 𝑎(⋅,⋅) is defined by 𝑎𝑢_ℎ,𝑢_ℎ=_Ω^𝑁_𝑖𝑗=1𝑎_𝑖𝑗(𝑥)𝜕𝑢^ℎ𝜕𝑥_𝑖𝜕𝑢_ℎ𝜕𝑥_𝑗+^𝑁_𝑗=1𝑏_𝑗(𝑥)𝜕𝑢^ℎ𝜕𝑥_𝑗𝑢_ℎ+𝑎₀(𝑥)𝑢^2ℎ𝑑𝑥,(2.9) under assumption (1.6), we have ^𝑁_𝑖𝑗=1_Ω𝑎_𝑖𝑗(𝑥)𝜕𝑢^ℎ𝜕𝑥_𝑖𝜕𝑢_ℎ𝜕𝑥_𝑗>𝛾^𝑁_𝑖=1_Ω𝜕𝑢_ℎ𝜕𝑥_𝑖²=𝛾‖‖∇𝑢_ℎ‖‖²²,(2.10) and since |||||_𝑁_𝑗=1_Ω𝑏_𝑗(𝑥)𝜕𝑢^ℎ𝜕𝑥_𝑗𝑢_ℎ|||||≤sup_𝑗||𝑏_𝑗||^𝑁_𝑗=1_Ω||||𝜕𝑢_ℎ𝜕𝑥_𝑗𝑢_ℎ≤‖‖𝑏_𝑗||||‖‖_∞‖‖∇𝑢_ℎ‖‖₂‖‖𝑢_ℎ‖‖₂,(2.11) then we make use of the algebraic inequality 𝑎𝑏≤12𝑎²+𝑏²,∀𝑎,𝑏∈ℝ,∀𝛾>0,(2.12) and choosing 𝑎𝑏==‖‖‖‖∇𝑏_𝑗𝑢_ℎ‖‖₂⋅√𝛾,‖‖_∞√𝛾‖‖𝑢^ℎ‖‖₂,(2.13) then we end up with |||||_𝑁_𝑗=1_Ω𝑏_𝑗(𝑥)𝜕𝑢^ℎ𝜕𝑥_𝑗𝑢_ℎ|||||≥−𝛾2‖‖∇𝑢^ℎ‖‖²²+‖‖𝑏^𝑗‖‖_∞⋅‖‖𝑢_ℎ‖‖₂2𝛾,(2.14) so we get 𝑎𝑢_ℎ,𝑢_ℎ≥𝛾‖‖∇𝑢_ℎ‖‖²²−𝛾2‖‖∇𝑢^ℎ‖‖²²+‖‖𝑏^𝑗‖‖_∞2𝛾‖‖𝑢^ℎ‖‖₂−‖‖𝑎₀‖‖_∞‖‖𝑢_ℎ‖‖²².(2.15) It can easily verified that 𝑎𝑢_ℎ,𝑢_ℎ≥𝛾2‖‖∇𝑢^ℎ‖‖₂+‖‖𝑢_ℎ‖‖²²−‖‖𝑎^𝑗‖‖^2∞2𝛾+𝛾2+‖‖𝑎⁰‖‖_∞‖‖𝑢^ℎ‖‖²².(2.16) Consequently, we deduce from above that 𝑎𝑢_ℎ,𝑢_ℎ+𝜆‖‖𝑢_ℎ‖‖²²≥𝛼‖‖𝑢_ℎ‖‖²¹suchthat𝛼=𝛾2,𝜆=‖‖𝑎^𝑗‖‖^2∞2𝛾+𝛾2+‖‖𝑎⁰‖‖_∞.(2.17)

We can identify the following result on the time energy behavior: 𝐸_ℎ(𝑡)=_Ω𝑢^2ℎ𝑑𝑥.(2.18)

Setting 𝑣=0 on (2.1) and after discretization by the finite element in the 𝑉^ℎ, we have the semidiscretization problem 𝜕𝜕𝑢𝑡^ℎ,𝑢^ℎ+𝑎𝑢^ℎ,𝑢_ℎ=12_Ω𝜕𝑢^2ℎ𝜕𝑡𝑑𝑥+𝑎𝑢^ℎ,𝑢_ℎ≤𝑓,𝑢_ℎ.(2.19)

Using Theorem 2.3, we deduce that12_Ω𝑑𝑢^2ℎ𝑑𝑡𝑑𝑥+𝑎𝑢^ℎ,𝑢_ℎ≥12𝑑𝑑𝑡_Ω𝑢^2ℎ𝑑𝑥+𝛼‖‖𝑢_ℎ‖‖²¹−𝜆‖‖𝑢_ℎ‖‖²²=12𝑑𝑑𝑡_Ω𝑢^2ℎ𝑑𝑥+𝛼‖‖𝑢_ℎ‖‖²²+𝛼‖‖∇𝑢_ℎ‖‖²²−𝜆‖‖𝑢_ℎ‖‖²²=12𝑑𝑑𝑡𝐸^ℎ≥1(𝑡)+2(𝛼−𝜆)𝐸_ℎ(𝑡)+2𝛼‖‖∇𝑢_ℎ‖‖²²𝑑𝑥2𝑑𝑑𝑡𝐸^ℎ(𝑡)+2(𝛼−𝜆)𝐸_ℎ(𝑡)+2𝛼𝐶^2∗_Ω𝑢^2ℎ𝑑𝑥.(2.20)

Thus, we have𝑑𝑑𝑡_Ω𝑢^2ℎ𝑑𝑥+𝛼‖‖𝑢_ℎ‖‖²¹−𝜆‖‖𝑢_ℎ‖‖²²≥𝑑𝑑𝑡𝐸^ℎ(𝑡)+2𝛼−𝜆+𝛼𝐶^2∗𝐸^ℎ(𝑡).(2.21)

Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find𝑓,𝑢_ℎ=_Ω𝑓(𝑥,𝑡)𝑢_ℎ(𝑥,𝑡)𝑑𝑥≤‖𝑓‖₂‖‖𝑢_ℎ‖‖₂.(2.22)

So that 𝑑𝑑𝑡𝐸^ℎ(𝑡)+2𝛼−𝜆+𝛼𝐶^2∗𝐸^ℎ(𝑡)≤2‖𝑓‖₂‖‖𝑢_ℎ‖‖₂.(2.23)

Using Young’s inequality𝑎𝑏≤𝜀𝑎²+14𝜀𝑏²,∀𝑎,𝑏∈ℝ,∀𝜀>0.(2.24)

Thus, we obtain𝑑𝑑𝑡𝐸^ℎ(𝑡)+2𝛼−𝜆+𝛼𝐶^2∗𝐸^ℎ(𝑡)≤2𝜀𝐸_ℎ(𝑡)+12𝜀‖𝑓‖²²,(2.25)

taking 𝜂=𝛼−𝜆+𝛼/𝐶^2∗, thus we have 𝑑𝑑𝑡𝐸^ℎ(𝑡)+2(𝜂−𝜀)𝐸_ℎ(𝑡)≤12𝜀‖𝑓‖²².(2.26)

Or, equivalently 𝑒^{2(𝜂−𝜀)𝑡}𝐸_ℎ(𝑡)^≤21𝜀𝑒^{2(𝜂−𝜀)𝑡}_Ω(𝑓(𝑥,𝑡))²𝑑𝑥.(2.27)

Integrating the last inequality from 0 to 𝑡, we get 𝐸_ℎ(𝑡)≤𝑒^{−2(𝜂−𝜀)𝑡}𝐸_ℎ(0)+12𝜀^𝑡0𝑒^{2(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 𝜕𝜕𝑢𝑡^ℎ,𝑣^ℎ−𝑢_ℎ+𝑎𝑢^ℎ,𝑣_ℎ−𝑢_ℎ≥𝑓,𝑣_ℎ−𝑢_ℎ,𝑣_ℎ∈𝑉^ℎ.(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 𝑢^𝑘ℎ−𝑢^ℎ𝑘−1,𝑣_ℎ−𝑢^ℎ𝜃,𝑘+Δ𝑡⋅𝑎𝑢^ℎ𝜃,𝑘,𝑣_ℎ−𝑢^ℎ𝜃,𝑘≥Δ𝑡⋅𝑓^𝜃,𝑘,𝑣_ℎ−𝑢^ℎ𝜃,𝑘,𝑣_ℎ∈𝑉^ℎ,(3.2) where 𝑓_𝜃,𝑘𝑢^ℎ𝜃,𝑘=𝜃𝑢^𝑘ℎ+(1−𝜃)𝑢^ℎ𝑘−1,=𝜃𝑓𝑡_𝑘+(1−𝜃)𝑓𝑡_𝑘−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 𝜔_𝑗∈𝐿²,𝜔_𝑗≠0:𝑎𝜔_𝑗,𝑣_ℎ=𝜆_𝑗𝜔_𝑗,𝑣_ℎ,∀𝑣_ℎ∈𝑉^ℎ.(3.4) The corresponding eigenfunctions {𝜔_𝑗} form a complete orthonormal basis in 𝐿²(Ω). In analogous way, when considering the finite dimensional problem in 𝑊^ℎ, we find a sequence of eigenvalues 𝛿≤𝜆_1ℎ≤𝜆_2ℎ≤⋯≤𝜆_𝑚(ℎ) and 𝐿²-orthonormal basis of eigenvectorss 𝜔_𝑖ℎ∈𝑊^ℎ,𝑖=1,2,…,𝑚(ℎ). Any function 𝑣_ℎ in 𝑉^ℎ can thus be expanded with respect to the system 𝜔_𝑖ℎ as 𝑣_ℎ=^𝑚(ℎ)_𝑖=1𝑣_ℎ,𝜔_𝑖ℎ𝜔_𝑖ℎ,(3.5) in particular, we have 𝑢^𝑘ℎ=^𝑚(ℎ)_𝑖=1𝑢^𝑘𝑖𝜔_𝑖ℎ,𝑢^𝑘𝑖=𝑢^𝑘ℎ,𝜔_𝑖ℎ.(3.6) Moreover, let 𝑓^𝑘ℎ be the 𝐿²-orthogonal projection of 𝑓^𝜃,𝑘 into 𝑊^ℎ, that is, 𝑓^𝑘ℎ∈𝑊^ℎ and 𝑓^𝑘ℎ,𝑣_ℎ=𝑓^𝜃,𝑘,𝑣_ℎ,(3.7) and set 𝑓^𝑘ℎ=^𝑚(ℎ)_𝑖=1𝑓^𝑘𝑖𝜔_𝑖ℎ,𝑓^𝑘𝑖=𝑓^𝑘ℎ,𝜔_𝑖ℎ.(3.8) We are now in a position to prove the stability for 𝜃∈[0,1/2[

Choosing in (3.1) 𝑣_ℎ=0, thus we have 1Δ𝑡𝑢^𝑘ℎ−𝑢^ℎ𝑘−1,𝑢^ℎ𝜃,𝑘+𝑎𝑢^ℎ𝜃,𝑘,𝑢^ℎ𝜃,𝑘≤𝑓^𝜃,𝑘,𝑢^ℎ𝜃,𝑘,𝑢^ℎ𝜃,𝑘∈𝑉^ℎ.(3.9)

The inequalities (3.2) is equivalent to 1Δ𝑡𝑢^𝑘𝑖−𝑢^𝑖𝑘−1+𝜆_𝑖ℎ𝜃𝑢^𝑘𝑖+(1−𝜃)𝑢^𝑖𝑘−1≤𝑓^𝑘𝑖.(3.10)

Since 𝜔_𝑖ℎ are the eigenfunctions means 𝑎𝜔_𝑖ℎ,𝜔_𝑖ℎ=𝜆_𝑖ℎ𝜔_𝑖ℎ,𝜔_𝑖ℎ=𝜆_𝑖ℎ⋅𝛿_𝑖𝑖=𝜆_𝑖ℎ,(3.11) for each 𝑘=0,…,𝑚(ℎ)−1, we can rewrite (3.9) as 𝑢^𝑘𝑖≤1−(1−𝜃)⋅Δ𝑡⋅𝜆^𝑖ℎ1+𝜃Δ𝑡⋅𝜆_𝑖ℎ𝑢^𝑖𝑘−1+Δ𝑡1+𝜃Δ𝑡⋅𝜆_𝑖ℎ𝑓^𝑘𝑖,(3.12) this inequality system stable if and only if ||||1−(1−𝜃)⋅Δ𝑡⋅𝜆_𝑖ℎ1+𝜃Δ𝑡⋅𝜆_𝑖ℎ||||<1,(3.13) that is to say 2𝜃−1>−2𝜆_𝑖ℎ⋅Δ𝑡(3.14) means Δ𝑡<2(1−2𝜃)𝜆_𝑖ℎ.(3.15)

So that this relation satisfied for all the eigenvalues 𝜆_𝑖ℎ of bilinear form 𝑎(⋅,⋅), we have to choose their highest value, and we take it for 𝜆_𝑚ℎ=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 Δ𝑡<2(1−2𝜃)𝜌(𝐴).(3.16)

We can prove that there exist two positive constants 𝑐₁,𝑐₂ such that 𝑐₁ℎ²≤𝜆_𝑚ℎ=𝑐₂ℎ⁻²,(3.17) thus the method of 𝜃-scheme is stable if and only if Δ𝑡<2𝐶(1−2𝜃)ℎ².(3.18)

Notice that this condition is always satisfied if 0≤𝜃<1/2. Hence, taking the absolute value of (3.12), we have ||𝑢^𝑚𝑖||<||𝑢^0𝑖||+Δ𝑡1+𝜃Δ𝑡⋅𝜆_{𝑖ℎ𝑚−1𝑖=1}𝑓^𝑘𝑖,(3.19) also we deduce that ‖‖𝑢^𝑚𝑖‖‖_∞<‖‖𝑢^0𝑖‖‖_∞+‖‖‖Δ𝑡1+𝜃Δ𝑡⋅𝜆_𝑖ℎ‖‖‖_∞𝑚−1_𝑖=1‖‖𝑓^𝑘𝑖‖‖_∞.(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 ‖‖𝑢^𝑘ℎ‖‖²²+2Δ𝑡^𝑛_𝑘=1𝑎𝑢^ℎ𝜃,𝑘,𝑢^ℎ𝜃,𝑘≤𝐶(𝑛)^𝑛_𝑘=1Δ𝑡‖‖𝑓^𝜃,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||logℎ||²,(4.1) where 𝑢^∞ and 𝑢^∞ℎ are, respectively, stationery solutions to the following continue and discrete inequalities: 𝑏𝑏(𝑢𝑢^∞,𝑣−𝑢^∞)≥(𝑓+𝜆𝑢^∞,𝑣−𝑢^∞),𝑣∈𝐻¹⁰(Ω),(4.2)_∞ℎ,𝑣_ℎ−𝑢^∞ℎ≥𝑓+𝜆𝑢^∞,𝑣_ℎ−𝑢^∞ℎ,𝑣_ℎ∈𝑉^ℎ,(4.3) such that 𝑏(⋅,⋅)=𝑎(⋅,⋅)+𝜆(⋅,⋅),(4.4)where 𝜆 is a positive constant arbitrary.We have 𝑢^ℎ𝜃,𝑘=𝜃𝑢^𝑘ℎ+(1−𝜃)𝑢^ℎ𝑘−1≤𝜃𝑟_ℎ𝜓+(1−𝜃)𝑟_ℎ𝜓=𝑟_ℎ𝜓.Thus, we can rewrite (3.1) as, for 𝑢^ℎ𝜃,𝑘∈𝑉^ℎ𝜃𝑢Δ^ℎ𝜃,𝑘𝑡,𝑣^ℎ−𝑢^𝑘ℎ̃+𝑎𝑢^ℎ𝜃,𝑘,𝑣_ℎ−𝑢^ℎ𝜃,𝑘≥𝑓^𝜃,𝑘+𝑢^ℎ𝑘−1𝜃Δ𝑡,𝑣^ℎ−𝑢^ℎ𝜃,𝑘,𝑣^ℎ∈𝑉^ℎ.(4.5) Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities: 𝑏𝑢^ℎ𝜃,𝑘,𝑣_ℎ−𝑢^𝑘ℎ̃≥𝑓^𝜃,𝑘+𝜇𝑢^ℎ𝑘−1,𝑣_ℎ−𝑢^ℎ𝜃,𝑘,𝑣_ℎ∈𝑉^ℎ,(4.6) such that 𝑏𝑢^ℎ𝜃,𝑘,𝑣_ℎ−𝑢^ℎ𝜃,𝑘=𝜇𝑢^ℎ𝜃,𝑘,𝑣_ℎ𝜇−=𝑢^ℎ𝜃,𝑘1+𝑎𝑢^ℎ𝜃,𝑘,𝑣_ℎ−𝑢^ℎ𝜃,𝑘,𝑣_ℎ,𝑢^ℎ𝜃,𝑘∈𝑉^ℎ,𝜃Δ𝑡=𝜃𝑇𝑘,(4.7) where 𝑢^ℎ𝜃,1 is the solution to the following discrete inequality: 𝑎𝑢^ℎ𝜃,1,𝑣_ℎ−𝑢^ℎ𝜃,1=𝑔𝑡_𝑘,𝑣_ℎ−𝑢^ℎ𝜃,1,𝑣_ℎ∈𝑉^ℎ,(4.8) where 𝑔(𝑡_𝑘) is a regular function given.

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

We consider the mapping 𝑇_ℎ∶𝐿^∞+(Ω)⟶𝑉^ℎ𝑤⟶𝑇_ℎ(𝑤)=𝜉_ℎ,(4.9) where 𝜉_ℎ is the unique solution of the following P.V.I: find 𝜉_ℎ∈𝑉^ℎ𝑏𝜉_ℎ,𝑣_ℎ−𝜉_ℎ≥𝑓^𝜃,𝑘+𝜇𝑤,𝑣_ℎ−𝜉_ℎ,𝑣_ℎ∈𝑉^ℎ.(4.10)

Proposition 4.2. Under the previous hypotheses and notations, if one sets 𝜃≥1/2, the mapping 𝑇_ℎ is a contraction in 𝐿^∞(Ω) with rate of contraction 1/(1+𝛽⋅𝜃⋅Δ𝑡). Therefore, 𝑇_ℎ 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_ℎ=𝑇_ℎ(𝑤)=𝜕(𝑓^𝜃,𝑘+𝜇𝑤,𝑟_ℎ𝜓) and 𝜉̃_ℎ=𝑇_ℎ(𝑤)=𝜕(𝑓^𝜃,𝑘+𝜇𝑤,𝑟_ℎ𝜓) solution to quasivariational inequalities (4.7) with right-hand side 𝐹^𝜃,𝑘=𝑓^𝜃,𝑘+𝜇𝑤_ℎ,𝐹^𝜃,𝑘=𝑓^𝜃,𝑘+𝜇𝑤_ℎ.
Now, setting 𝜙=1𝛽+𝜇‖‖𝐹^𝜃,𝑘−𝐹^𝜃,𝑘‖‖_∞,(4.11) then for 𝜉_ℎ+𝜙 is solution of 𝑏𝜉_ℎ+𝜙,𝜉𝑣+_ℎ𝜙+≤𝜙𝑟_ℎ−𝜉_ℎ+𝜙≥𝐹^𝜃,𝑘+𝑎₀𝜙,𝑣_ℎ+𝜙−𝜉_ℎ+𝜙,𝜓+𝜙,𝑣_ℎ+𝜙≤𝑟_ℎ𝜓+𝜙,∀𝑣_ℎ∈𝑉^ℎ.(4.12) Also, we have 𝐹^𝜃,𝑘≤𝐹^𝜃,𝑘+‖‖𝐹^𝜃,𝑘−𝐹^𝜃,𝑘‖‖_∞≤𝐹^𝜃,𝑘+𝑎⁰𝛽+𝜇‖‖𝐹^𝜃,𝑘−𝐹^𝜃,𝑘‖‖_∞≤𝐹^𝜃,𝑘+𝑎₀𝜙,(4.13) thus 𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓+𝜙≤𝜕_ℎ𝐹^𝜃,𝑘+𝑎₀(𝑥)𝜙,𝑟_ℎ𝜓+𝜙≤𝜕_ℎ𝐹^𝜃,𝑘,𝑟𝜓+𝜙,(4.14) hence 𝜉_ℎ≤𝜉̃_ℎ+𝜙.(4.15) Similarly, interchanging the roles of 𝑤 and 𝑤, we also get 𝜉̃_ℎ≤𝜉_ℎ+𝜙.(4.16) Finally, this yields ‖‖𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓−𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓‖‖_∞≤𝛽+1𝜇‖‖𝐹^𝜃,𝑘−𝐹^𝜃,𝑘‖‖_∞≤1𝛽+𝜇‖‖𝑓^𝜃,𝑘+𝜇𝑤−𝑓^𝜃,𝑘−𝜇𝑤‖‖_∞‖‖𝑤−𝑤‖‖_∞≤11+𝛽𝜃Δ𝑡‖‖𝑤−𝑤‖‖^∞,(4.17) which completes the proof.

Remark 4.3. If we set 0≤𝜃<1/2, the mapping 𝑇_ℎ 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 ‖‖𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓−𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓‖‖_∞≤1+1𝛽𝜃Δ𝑡‖‖𝑤−𝑤‖‖^∞≤22+𝛽𝜃(1−2𝜃)𝜌(𝐴)‖‖𝑤−𝑤‖‖^∞≤11+𝛽𝜃(1−2𝜃)/2𝐶ℎ²‖‖𝑤−𝑤‖‖^∞=2𝐶ℎ²2𝐶ℎ²+𝛽𝜃(1−2𝜃)‖‖𝑤−𝑤‖‖^∞,(4.18) also it can be found that ‖‖𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓−𝜕_ℎ𝐹^𝜃,𝑘,𝑟_ℎ𝜓‖‖_∞≤11+𝛽𝜃(1−2𝜃)/2𝐶ℎ²‖‖𝑤−𝑤‖‖^∞=2𝐶ℎ²2𝐶ℎ²+𝛽𝜃(1−2𝜃)‖‖𝑤−𝑤‖‖^∞,(4.19) thus the mapping 𝑇_ℎ is a contraction in 𝐿^∞(Ω) with rate of contraction (2𝐶ℎ²)/(2𝐶ℎ²+𝛽𝜃(1−2𝜃)). Therefore, 𝑇_ℎ admits a unique fixed point which coincides with the solution of P.V.I (4.7) ‖‖𝑇_ℎ(𝑤)−T_ℎ𝑤‖‖_∞≤1𝛽+𝜇‖‖𝐹^𝜃,𝑘−𝐹^𝜃,𝑘‖‖_∞=1𝛽+𝜇‖‖𝑓^𝜃,𝑘+𝜇𝑤−𝑓^𝜃,𝑘−𝜇𝑤‖‖_∞≤𝜇𝛽+𝜇‖‖𝑤−𝑤‖‖^∞≤11+𝛽𝜃Δ𝑡‖‖𝑤−𝑤‖‖^∞.(4.20) This completes the proof.

4.2. Discrete Algorithm

Starting from 𝑢^0ℎ=𝑢_0ℎ (initial data) and the 𝑢^ℎ𝜃,1 solution of problem (4.7), we introduce the following discrete algorithm: 𝑢^ℎ𝜃,𝑘=𝑇_ℎ𝑢^ℎ𝑘−1,𝑘=1,…,𝑛,(4.21) where 𝑢^ℎ𝜃,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‖‖𝑢^ℎ𝜃,𝑘−𝑢^∞ℎ‖‖_∞≤11+𝛽𝜃Δ𝑡^𝑘‖‖𝑢^∞ℎ−𝑢_ℎ0‖‖_∞,(4.22) and one has for ‖‖𝑢^ℎ𝜃,𝑘−𝑢^∞ℎ‖‖_∞≤2𝐶ℎ²2𝐶ℎ²+𝛽𝜃(1−2𝜃)^𝑘‖‖𝑢^∞ℎ−𝑢_ℎ0‖‖_∞for0≤𝜃<12.(4.23)

Proof. we set a first case 𝜃≥1/2, and we have ‖‖𝑢_ℎ𝜃,1𝑢^∞ℎ=𝑇_ℎ𝑢^∞ℎ,−𝑢^∞ℎ‖‖_∞=‖‖𝑇_ℎ𝑢^0ℎ−𝑇_ℎ𝑢^∞ℎ‖‖_∞≤11+𝛽𝜃Δ𝑡‖‖𝑢^0ℎ−𝑢^∞ℎ‖‖_∞.(4.24) for 𝑘≥2, we use the Bensoussan-Lions' algorithm (𝑢^𝑘ℎ=𝑇_ℎ𝑢^ℎ𝑘−1,𝑘=1,…,𝑛) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that ‖‖𝑢^ℎ𝜃,𝑘−𝑢^∞ℎ‖‖_∞≤11+𝛽𝜃Δ𝑡^𝑘‖‖𝑢^0ℎ−𝑢^∞ℎ‖‖_∞,(4.25) so ‖‖𝑢^{ℎ𝜃,𝑘+1}−𝑢^∞ℎ‖‖_∞=‖‖𝑇_ℎ𝑢^𝑘ℎ−𝑇_ℎ𝑢^∞ℎ‖‖_∞≤11+𝛽𝜃Δ𝑡‖‖𝑢^𝑘ℎ−𝑢^∞ℎ‖‖_∞,(4.26) thus ‖‖𝑢^{ℎ𝜃,𝑘+1}−𝑢^∞ℎ‖‖_∞≤11+𝛽𝜃Δ𝑡^𝑘+1‖‖𝑢^0ℎ−𝑢^∞ℎ‖‖_∞,(4.27) for a second case 0≤𝜃<1/2, it can be easily shown that ‖‖𝑢^ℎ𝜃,𝑘−𝑢^∞ℎ‖‖_∞≤2𝐶ℎ²2𝐶ℎ²+𝛽𝜃(1−2𝜃)^𝑘‖‖𝑢^0ℎ−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 𝑢^𝜃ℎ(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, ‖‖𝑢^ℎ𝜃,𝑛−𝑢^∞‖‖_∞≤𝐶ℎ²||logℎ||²+11+𝛽𝜃Δ𝑡^𝑛,(4.29) and for the second case 0≤𝜃<1/2, ‖‖𝑢^ℎ𝜃,𝑛−𝑢^∞‖‖_∞≤𝐶ℎ²||logℎ||²+2𝐶ℎ²2𝐶ℎ²+𝛽𝜃(1−2𝜃)^𝑛,(4.30) where 𝐶 is a constant independent of ℎ and 𝑘.

Proof. We have 𝑢^ℎ𝜃,𝑘(𝑥)=𝑢_ℎ(𝑡,𝑥)for𝑡∈](𝑘−1)Δ𝑡;𝑘Δ𝑡[,(4.31) thus 𝑢^ℎ𝜃,𝑛(𝑥)=𝑢_ℎ(𝑇,𝑥),(4.32) then ‖‖𝑢^𝜃ℎ(𝑇,𝑥)−𝑢^∞‖‖_𝐿∞(Ω)=‖‖𝑢^ℎ𝜃,𝑛−𝑢^∞‖‖_𝐿∞(Ω)≤‖‖𝑢^ℎ𝜃,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, ‖‖𝑢^ℎ𝜃,𝑛−𝑢^∞‖‖_∞≤𝐶ℎ²||logℎ||²+11+𝛽𝜃Δ𝑡^𝑛,(4.34) and for 0≤𝜃<1/2, we have ‖‖𝑢^ℎ𝜃,𝑛−𝑢^∞‖‖_∞≤𝐶ℎ²||logℎ||²+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.