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

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.


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:

ISRN Mathematical Analysis
We consider the following problem, find u ∈ K solution of 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 L 2 Ω , and A is an operator defined over H 1 Ω by b j x ∂u ∂x j a 0 x u, 1.4 and whose coefficients: a i,j x , b j x , a 0 x ∈ L ∞ Ω ∩ C 2 Ω , x ∈ Ω, 1 ≤ i, j ≤ N are sufficiently smooth functions and satisfy the following conditions: f is a regular functions satisfying We specify the following notations: 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 H 1 0 by a space discretization of finite dimensional V h ⊂ H 1 0 .In the second step, we discretize the problem with respect to time using the θ-scheme.Therefore, we search a sequence of elements u n h ∈ V h which approaches u n t n , t n nΔt, with initial data u 0 h 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.

Priory Estimate of the Discrete Parabolic Variational Inequalities
We can reformulate 1.3 to the following variational inequality: where a •, • is the bilinear form associated with operator A defined in 1.4 .Namely,

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 quasiuniform, 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:

2.5
We consider r h to be the usual interpolation operator defined by 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.

Priory Estimate
Theorem 2.3.Let us assume that the discrete bilinear form a •, • defined as 2.2 is weakly coercive in V h ⊂ H 1 0 Ω .Then, there exists two constants α > 0 and λ > 0 such that Proof.The bilinear form a •, • is defined by under assumption 1.6 , we have and since

2.11
then we make use of the algebraic inequality 12 and choosing

2.13
then we end up with

2.15
It can easily verified that

2.16
Consequently, we deduce from above that

2.17
We can identify the following result on the time energy behavior:

2.18
Setting v 0 on 2.1 and after discretization by the finite element in the V h , we have the semidiscretization problem

2.19
Using Theorem 2.3, we deduce that

ISRN Mathematical Analysis
Thus, we have

2.21
Applying the Cauchy-Schwartz inequality on the right-hand side of 2.1 , we find

2.23
Using Young's inequality Thus, we obtain

2.27
Integrating the last inequality from 0 to t, we get

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.

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 Now, we apply the θ-scheme on the semidiscrete problem 3.1 ; for any θ ∈ 0, 1 and k 1, . . ., n, we have where

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 L 2 Ω , since Ω is bounded.Thus, there exists a nondecreasing sequence of eigenvalues δ ≤ λ 1 ≤ λ 2 ≤ • • • for the bilinear form a •, • satisfying ω j ∈ L 2 , 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 L 2 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 L 2 -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 in particular, we have

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 3.9 The inequalities 3.2 is equivalent to

3.10
Since ω ih are the eigenfunctions means for each k 0, . . ., m h − 1, we can rewrite 3.9 as 3.12 this inequality system stable if and only if

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

3.16
We can prove that there exist two positive constants c 1 , c 2 such that 3.17 thus the method of θ-scheme is stable if and only if

3.18
Notice that this condition is always satisfied if 0 ≤ θ < 1/2.Hence, taking the absolute value of 3.12 , we have also we deduce that

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.

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 where u ∞ and u ∞ h are, respectively, stationery solutions to the following continue and discrete inequalities: where λ is a positive constant arbitrary.We have Thus, our problem 4.5 is equivalent to the following noncoercive elliptic variational inequalities:

4.7
where u θ,1 h is the solution to the following discrete inequality: where g t k is a regular function given.

A Fixed Point Mapping Associated with Discrete Problem 4.7
We consider the mapping

4.9
where ξ h is the unique solution of the following P.V.I: find 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 h ψ and ξ h T h w ∂ f θ,k μ w, r h ψ solution to quasivariational inequalities 4.7 with right-hand side

4.12
Also, we have Similarly, interchanging the roles of w and w, we also get Finally, this yields 4.18 also it can be found that thus the mapping T h is a contraction in L ∞ Ω with rate of contraction 2Ch 2 / 2Ch 2 βθ 1 − 2θ .Therefore, T h admits a unique fixed point which coincides with the solution of P.V.I 4.7

4.20
This completes the proof.

Discrete Algorithm
Starting from u 0 h u 0h initial data and the u θ,1 h solution of problem 4.7 , we introduce the following discrete algorithm: and one has for Proof.we set a first case θ ≥ 1/2, and we have
We assume that ISRN Mathematical Analysis for a second case 0 ≤ θ < 1/2, it can be easily shown that

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 θ h T, x , the discrete solution calculated at the moment T nΔt and u ∞ , the asymptotic continuous solution of 4.

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.
4.1 and Proposition 4.5, we have for θ ≥ 1/2,u θ,n h − u ∞ ∞ ≤ C h 2 log h Sobolev-Poincare inequality .Let Ω be a bounded overt in R N , with sufficiently smooth boundary Γ, then there exists a C * such that 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 h 2 .Thus it can be easily show that 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 .Under the previous hypotheses, one has the following estimate of convergence:

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, n , 4.30where C is a constant independent of h and k.