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 time scheme combined with a finite element spatial approximation for parabolic inequalities.

Let us assume that is an implicit convex set defined as follows: with

We consider the following problem, find solution of where is a set in defined as with , and is convex domain in , with sufficiently smooth boundary .

The symbol stands for the inner product in , and is an operator defined over by and whose coefficients: are sufficiently smooth functions and satisfy the following conditions: 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 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 by a space discretization of finite dimensional . In the second step, we discretize the problem with respect to time using the -scheme. Therefore, we search a sequence of elements which approaches , with initial data . 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 -stability analysis of the -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 -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: where is the bilinear form associated with operator defined in (1.4). Namely,

Theorem 2.1 (see [9]). The problem (1.3) has an unique solution . Moreover, one has

Lemma 2.2 (Sobolev-Poincare inequality). Let be a bounded overt in , with sufficiently smooth boundary , then there exists a such that

2.1. The Discrete Problem

Let us assume that can be decomposed into triangles and denotes the set of all the elements , where is the mesh size. We assume that the family is regular and quasi-uniform, and we consider the usual basis of affine functions , defined by , where is a vertex of the considered triangulation. We introduce the following discrete spaces of finite element:

We consider to be the usual interpolation operator defined by

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

2.1.1. Priory Estimate

Theorem 2.3. Let us assume that the discrete bilinear form defined as (2.2) is weakly coercive in . Then, there exists two constants and such that where

Proof. The bilinear form is defined by under assumption (1.6), we have and since then we make use of the algebraic inequality and choosing then we end up with so we get It can easily verified that Consequently, we deduce from above that

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

Setting on (2.1) and after discretization by the finite element in the , we have the semidiscretization problem

Using Theorem 2.3, we deduce that

Thus, we have

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

So that

Using Young’s inequality

Thus, we obtain

taking , thus we have

Or, equivalently

Integrating the last inequality from 0 to , we get

Remark 2.4. In particular, when and choosing , then (2.28) shows that the energy decreasing exponentially fast in time.

3. The -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

Now, we apply the -scheme on the semidiscrete problem (3.1); for any and , we have where

It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear form . We recall that is compactly embedded in , since is bounded. Thus, there exists a nondecreasing sequence of eigenvalues for the bilinear form satisfying The corresponding eigenfunctions form a complete orthonormal basis in . In analogous way, when considering the finite dimensional problem in , we find a sequence of eigenvalues and -orthonormal basis of eigenvectorss . Any function in can thus be expanded with respect to the system as in particular, we have Moreover, let be the -orthogonal projection of into , that is, and and set We are now in a position to prove the stability for

Choosing in (3.1) , thus we have

The inequalities (3.2) is equivalent to

Since are the eigenfunctions means for each , we can rewrite (3.9) as this inequality system stable if and only if that is to say means

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

We deduce that if the -scheme way is stable unconditionally (i.e., stable for all Δt). However, if the -scheme is stable unless

We can prove that there exist two positive constants such that thus the method of -scheme is stable if and only if

Notice that this condition is always satisfied if . Hence, taking the absolute value of (3.12), we have also we deduce that

Remark 3.1 (cf. [4]). We assume that the coerciveness assumption (Theorem 2.3) is satisfied with , and for each , we find

4. Asymptotic Behavior of -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 independent of such that where and are, respectively, stationery solutions to the following continue and discrete inequalities: such that where is a positive constant arbitrary.We have .Thus, we can rewrite (3.1) as, for Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities: such that where is the solution to the following discrete inequality: where is a regular function given.

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

We consider the mapping where is the unique solution of the following P.V.I: find

Proposition 4.2. Under the previous hypotheses and notations, if one sets , the mapping is a contraction in with rate of contraction . Therefore, admits a unique fixed point which coincides with the solution of P.V. I (4.7).

Proof. For , in , we consider and solution to quasivariational inequalities (4.7) with right-hand side .
Now, setting then for is solution of Also, we have thus hence Similarly, interchanging the roles of and , we also get Finally, this yields which completes the proof.

Remark 4.3. If we set , the mapping is a contraction in with rate of contraction , where is a spectral radius of operator .

Proof. Under condition of stability, we have shown the -scheme is stable if and only if .
Thus it can be easily show that also it can be found that thus the mapping is a contraction in with rate of contraction . Therefore, admits a unique fixed point which coincides with the solution of P.V.I (4.7) This completes the proof.

4.2. Discrete Algorithm

Starting from (initial data) and the solution of problem (4.7), we introduce the following discrete algorithm: where is the solution of the problem (4.7).

Remark 4.4. If we choose 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 and one has for

Proof. we set a first case , and we have for , we use the Bensoussan-Lions' algorithm for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that so thus for a second case , it can be easily shown that