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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.

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., [

In this paper, we propose a new iterative discrete algorithm to prove the existence and uniqueness, and we devote the asymptotic behavior using the

Let us assume that

We consider the following problem, find

The symbol

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

The paper is organized as follows. In Section

We can reformulate (

The problem (

Let

Let us assume that

We consider

The matrix whose coefficients

Let us assume that the discrete bilinear form

The bilinear form

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

Setting

Using Theorem

Thus, we have

Applying the Cauchy-Schwartz inequality on the right-hand side of (

So that

Using Young’s inequality

Thus, we obtain

taking

Or, equivalently

Integrating the last inequality from 0 to

In particular, when

We apply the finite element method to approximate inequality (

Now, we apply the

It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear form

Choosing in (

The inequalities (

Since

So that this relation satisfied for all the eigenvalues

We deduce that if

We can prove that there exist two positive constants

Notice that this condition is always satisfied if

We assume that the coerciveness assumption (Theorem

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.

Under the previous assumptions, and the maximum principle, there exists a constant

We consider the mapping

Under the previous hypotheses and notations, if one sets

For

Now, setting

If we set

Under condition of stability, we have shown the

Thus it can be easily show that

Starting from

If we choose

Under the previous hypotheses, one has the following estimate of convergence: if

we set a first case

We assume that

This section is devoted to the proof of main result of the present paper, where we prove the theorem of the asymptotic behavior in

Now, we evaluate the variation in

Under condition of Theorem

We have

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.

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