International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 703670 | https://doi.org/10.5402/2011/703670

Salah Boulaaras, Mohamed Haiour, "A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities", International Scholarly Research Notices, vol. 2011, Article ID 703670, 15 pages, 2011. https://doi.org/10.5402/2011/703670

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

Academic Editor: S. Zhang
Received30 Mar 2011
Accepted13 May 2011
Published07 Jul 2011

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., [1โ€“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 ๐œƒ time scheme combined with a finite element spatial approximation for parabolic inequalities.

Let us assume that ๐พ is an implicit convex set defined as follows: ๎€ฝ๐พ=๐‘ฃ(๐‘ก,๐‘ฅ)โˆˆ๐ฟ2๎€ท0,๐‘‡,๐ป10๎€ธ(ฮฉ),๐‘ฃ(๐‘ก,๐‘ฅ)โ‰ค๐œ“(๐‘ก,๐‘ฅ),๐‘ฃ(0,๐‘ฅ)=๐‘ฃ0๎€พinฮฉ,(1.1) with ๐œ“โˆˆ๐ฟ2๎€ท0,๐‘‡,๐‘Š2,โˆž๎€ธ(ฮฉ).(1.2)

We consider the following problem, find ๐‘ขโˆˆ๐พ solution of ๐œ•๐‘ข๐œ•๐‘ก+๐ด๐‘ขโ‰ค๐‘“inฮฃ,๐‘ข(๐‘ก,๐‘ฅ)=0inฮ“,(1.3) where ฮฃ is a set in โ„ร—โ„๐‘ defined as ฮฃ=ฮฉร—[0,๐‘‡] with ๐‘‡โ‹…<+โˆž, and ฮฉ is convex domain in โ„๐‘, with sufficiently smooth boundary ฮ“.

The symbol (โ‹…,โ‹…) stands for the inner product in ๐ฟ2(ฮฉ), and ๐ด is an operator defined over ๐ป1(ฮฉ) by ๐ด๐‘ข=โˆ’๐‘๎“๐‘–๐‘—=1๐œ•๐œ•๐‘ฅ๐‘–๐‘Ž๐‘–๐‘—(๐‘ฅ)๐œ•๐‘ข๐œ•๐‘ฅ๐‘—+๐‘๎“๐‘—=1๐‘๐‘—(๐‘ฅ)๐œ•๐‘ข๐œ•๐‘ฅ๐‘—+๐‘Ž0(๐‘ฅ)๐‘ข,(1.4) and whose coefficients: ๐‘Ž๐‘–,๐‘—(๐‘ฅ),๐‘๐‘—(๐‘ฅ),๐‘Ž0(๐‘ฅ)โˆˆ๐ฟโˆž(ฮฉ)โˆฉ๐ถ2(ฮฉ),๐‘ฅโˆˆฮฉ,1โ‰ค๐‘–,๐‘—โ‰ค๐‘ are sufficiently smooth functions and satisfy the following conditions: ๐‘Ž๐‘–๐‘—(๐‘ฅ)=๐‘Ž๐‘—๐‘–(๐‘ฅ);๐‘Ž0(๐‘ฅ)โ‰ฅ๐›ฝ>0,๐›ฝisaconstant,(1.5)๐‘๎“๐‘–๐‘—=1๐‘Ž๐‘–๐‘—(๐‘ฅ)๐œ‰๐‘–๐œ‰๐‘—||๐œ‰||โ‰ฅ๐›พ2;๐œ‰โˆˆโ„๐‘,๐›พ>0,๐‘ฅโˆˆฮฉ,(1.6)๐‘“ is a regular functions satisfying ๐‘“โˆˆ๐ฟ2(0,๐‘‡,๐ฟโˆž(ฮฉ))โˆฉ๐ถ1๎€ท0,๐‘‡,๐ปโˆ’1๎€ธ(ฮฉ),๐‘“โ‰ฅ0.(1.7)

We specify the following notations: โ€–โ‹…โ€–๐ฟ2(ฮฉ)=โ€–โ‹…โ€–2,โ€–โ‹…โ€–1=โ€–โ‹…โ€–๐ป10(ฮฉ),โ€–โ‹…โ€–๐ฟโˆž(ฮฉ)=โ€–โ‹…โ€–โˆž.(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 ๐œƒ 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 ๐ป10 by a space discretization of finite dimensional ๐‘‰โ„ŽโŠ‚๐ป10. In the second step, we discretize the problem with respect to time using the ๐œƒ-scheme. Therefore, we search a sequence of elements ๐‘ข๐‘›โ„Žโˆˆ๐‘‰โ„Ž which approaches ๐‘ข๐‘›(๐‘ก๐‘›),๐‘ก๐‘›=๐‘›ฮ”๐‘ก, with initial data ๐‘ข0โ„Ž=๐‘ข0โ„Ž. 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 ๐ฟโˆž-stability analysis of the ๐œƒ-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 ๐œƒ-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: ๎‚€๐œ•๐‘ข๎‚๐œ•๐‘ก,๐‘ฃโˆ’๐‘ข+๐‘Ž(๐‘ข,๐‘ฃโˆ’๐‘ข)โ‰ฅ(๐‘“,๐‘ฃโˆ’๐‘ข),๐‘ฃโˆˆ๐พ,(2.1) where ๐‘Ž(โ‹…,โ‹…) is the bilinear form associated with operator ๐ด defined in (1.4). Namely, ๎€œ๐‘Ž(๐‘ข,๐‘ฃ)=ฮฉ๎ƒฉ๐‘๎“๐‘–๐‘—=1๐‘Ž๐‘–๐‘—(๐‘ฅ)๐œ•๐‘ข๐œ•๐‘ฅ๐‘–๐œ•๐‘ฃ๐œ•๐‘ฅ๐‘—+๐‘๎“๐‘—=1๐‘๐‘—(๐‘ฅ)๐œ•๐‘ข๐œ•๐‘ฅ๐‘—๐‘ฃ+๐‘Ž0๎ƒช(๐‘ฅ)๐‘ข๐‘ฃ๐‘‘๐‘ฅ,(2.2)

Theorem 2.1 (see [9]). The problem (1.3) has an unique solution ๐‘ขโˆˆ๐พ(๐‘ข). Moreover, one has ๐‘ขโˆˆ๐ฟ2๎€ท0,๐‘‡;๐ป10๎€ธ,(ฮฉ)๐œ•๐‘ข๐œ•๐‘กโˆˆ๐ฟ2๎€ท0,๐‘‡;๐ปโˆ’1๎€ธ(ฮฉ).(2.3)

Lemma 2.2 (Sobolev-Poincare inequality). Let ฮฉ be a bounded overt in โ„๐‘, with sufficiently smooth boundary ฮ“, then there exists a ๐ถโˆ— such that โ€–๐‘ขโ€–2โ‰ค๐ถโˆ—โ€–โˆ‡๐‘ขโ€–2,๐‘ฃโˆˆ๐ป10(ฮฉ)โˆฉ๐ถ2๎‚€ฮฉ๎‚,โˆ‡=๐‘๎“๐‘–=1๐œ•๐œ•๐‘ฅ๐‘–.(2.4)

2.1. The Discrete Problem

Let us assume that ฮฉ can be decomposed into triangles and ๐œโ„Ž denotes the set of all the elements โ„Ž>0, where โ„Ž is the mesh size. We assume that the family ๐œโ„Ž is regular and quasi-uniform, and we consider the usual basis of affine functions ๐œ‘๐‘–, ๐‘–={1,โ€ฆ,๐‘š(โ„Ž)} defined by ๐œ‘๐‘–(๐‘€๐‘—)=๐›ฟ๐‘–๐‘—, where ๐‘€๐‘— is a vertex of the considered triangulation. We introduce the following discrete spaces ๐‘‰โ„Ž of finite element: ๐‘‰โ„Ž=๎‚†๐‘ฃโ„Žโˆˆ๐ฟ2๎€ท0,๐‘‡,๐ป10๎€ธ๎‚€(ฮฉ)โˆฉ๐ถ0,๐‘‡,๐ป10๎‚€ฮฉ๎‚๎‚,suchthat๐‘ฃโ„Ž|๐‘˜โˆˆ๐‘ƒ1,๐‘˜โˆˆ๐œโ„Ž,๐‘ฃโ„Žโ‰ค๐‘Ÿโ„Ž๐œ“,๐‘ฃโ„Ž(โ‹…,0)=๐‘ฃโ„Ž0๎‚‡.inฮฉ.(2.5)

We consider ๐‘Ÿโ„Ž to be the usual interpolation operator defined by ๐‘ฃโˆˆ๐ฟ2๎€ท0,๐‘‡,๐ป10๎€ธ๎‚€(ฮฉ)โˆฉ๐ถ0,๐‘‡,๐ป10๎‚€ฮฉ๎‚๎‚,๐‘Ÿโ„Ž๐‘ฃ=๐‘š(โ„Ž)๎“๐‘–=1๐‘ฃ๎€ท๐‘€๐‘–๎€ธ๐œ‘๐‘–(๐‘ฅ).(2.6)

The Discrete Maximum Principle Assumption (see [10])
The matrix whose coefficients ๐‘Ž(๐œ‘๐‘–,๐œ‘๐‘—) are supposed to be M-matrix. For convenience, in all the sequels, ๐ถ will be a generic constant independent on โ„Ž.

2.1.1. Priory Estimate

Theorem 2.3. Let us assume that the discrete bilinear form ๐‘Ž(โ‹…,โ‹…) defined as (2.2) is weakly coercive in ๐‘‰โ„ŽโŠ‚๐ป10(ฮฉ). Then, there exists two constants ๐›ผ>0 and ๐œ†>0 such that ๐‘Ž๎€ท๐‘ขโ„Ž,๐‘ขโ„Ž๎€ธโ€–โ€–๐‘ข+๐œ†โ„Žโ€–โ€–2โ€–โ€–๐‘ขโ‰ฅ๐›ผโ„Žโ€–โ€–1,(2.7) where ๎ƒฉโ€–โ€–๐‘๐œ†=๐‘—โ€–โ€–2โˆž+๐›พ2๐›พ2+โ€–โ€–๐‘Ž0โ€–โ€–โˆž๎ƒช๐›พ,๐›ผ=2.(2.8)

Proof. The bilinear form ๐‘Ž(โ‹…,โ‹…) is defined by ๐‘Ž๎€ท๐‘ขโ„Ž,๐‘ขโ„Ž๎€ธ=๎€œฮฉ๎ƒฉ๐‘๎“๐‘–๐‘—=1๐‘Ž๐‘–๐‘—(๐‘ฅ)๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘–๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘—+๐‘๎“๐‘—=1๐‘๐‘—(๐‘ฅ)๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘—๐‘ขโ„Ž+๐‘Ž0(๐‘ฅ)๐‘ข2โ„Ž๎ƒช๐‘‘๐‘ฅ,(2.9) under assumption (1.6), we have ๐‘๎“๐‘–๐‘—=1๎€œฮฉ๐‘Ž๐‘–๐‘—(๐‘ฅ)๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘–๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘—>๐›พ๐‘๎“๐‘–=1๎‚ต๎€œฮฉ๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘–๎‚ถ2โ€–โ€–=๐›พโˆ‡๐‘ขโ„Žโ€–โ€–22,(2.10) and since |||||๐‘๎“๐‘—=1๎€œฮฉ๐‘๐‘—(๐‘ฅ)๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘—๐‘ขโ„Ž|||||โ‰คsup๐‘—||๐‘๐‘—||๐‘๎“๐‘—=1๎€œฮฉ||||๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘—๐‘ขโ„Ž||||โ‰คโ€–โ€–๐‘๐‘—โ€–โ€–โˆžโ€–โ€–โˆ‡๐‘ขโ„Žโ€–โ€–2โ€–โ€–๐‘ขโ„Žโ€–โ€–2,(2.11) then we make use of the algebraic inequality 1๐‘Ž๐‘โ‰ค2๎€ท๐‘Ž2+๐‘2๎€ธ,โˆ€๐‘Ž,๐‘โˆˆโ„,โˆ€๐›พ>0,(2.12) and choosing โ€–โ€–๐‘Ž=โˆ‡๐‘ขโ„Žโ€–โ€–2โ‹…โˆšโ€–โ€–๐‘๐›พ,๐‘=๐‘—โ€–โ€–โˆžโˆš๐›พโ€–โ€–๐‘ขโ„Žโ€–โ€–2,(2.13) then we end up with |||||๐‘๎“๐‘—=1๎€œฮฉ๐‘๐‘—(๐‘ฅ)๐œ•๐‘ขโ„Ž๐œ•๐‘ฅ๐‘—๐‘ขโ„Ž|||||๎ƒฉ๐›พโ‰ฅโˆ’2โ€–โ€–โˆ‡๐‘ขโ„Žโ€–โ€–22+โ€–โ€–๐‘๐‘—โ€–โ€–โˆžโ‹…โ€–โ€–๐‘ขโ„Žโ€–โ€–2๎ƒช2๐›พ,(2.14) so we get ๐‘Ž๎€ท๐‘ขโ„Ž,๐‘ขโ„Ž๎€ธโ€–โ€–โ‰ฅ๐›พโˆ‡๐‘ขโ„Žโ€–โ€–22โˆ’๐›พ2โ€–โ€–โˆ‡๐‘ขโ„Žโ€–โ€–22+โ€–โ€–๐‘๐‘—โ€–โ€–โˆžโ€–โ€–๐‘ข2๐›พโ„Žโ€–โ€–2โˆ’โ€–โ€–๐‘Ž0โ€–โ€–โˆžโ€–โ€–๐‘ขโ„Žโ€–โ€–22.(2.15) It can easily verified that ๐‘Ž๎€ท๐‘ขโ„Ž,๐‘ขโ„Ž๎€ธโ‰ฅ๐›พ2๎‚€โ€–โ€–โˆ‡๐‘ขโ„Žโ€–โ€–2+โ€–โ€–๐‘ขโ„Žโ€–โ€–22๎‚โˆ’๎ƒฉโ€–โ€–๐‘Ž๐‘—โ€–โ€–2โˆž+๐›พ2๐›พ2+โ€–โ€–๐‘Ž0โ€–โ€–โˆž๎ƒชโ€–โ€–๐‘ขโ„Žโ€–โ€–22.(2.16) Consequently, we deduce from above that ๐‘Ž๎€ท๐‘ขโ„Ž,๐‘ขโ„Ž๎€ธโ€–โ€–๐‘ข+๐œ†โ„Žโ€–โ€–22โ€–โ€–๐‘ขโ‰ฅ๐›ผโ„Žโ€–โ€–21๐›พsuchthat๐›ผ=2๎ƒฉโ€–โ€–๐‘Ž,๐œ†=๐‘—โ€–โ€–2โˆž+๐›พ2๐›พ2+โ€–โ€–๐‘Ž0โ€–โ€–โˆž๎ƒช.(2.17)

We can identify the following result on the time energy behavior: ๐ธโ„Ž(๎€œ๐‘ก)=ฮฉ๐‘ข2โ„Ž๐‘‘๐‘ฅ.(2.18)

Setting ๐‘ฃ=0 on (2.1) and after discretization by the finite element in the ๐‘‰โ„Ž, we have the semidiscretization problem ๎‚ต๐œ•๐‘ขโ„Ž๐œ•๐‘ก,๐‘ขโ„Ž๎‚ถ๎€ท๐‘ข+๐‘Žโ„Ž,๐‘ขโ„Ž๎€ธ=12๎€œฮฉ๐œ•๐‘ข2โ„Ž๎€ท๐‘ข๐œ•๐‘ก๐‘‘๐‘ฅ+๐‘Žโ„Ž,๐‘ขโ„Ž๎€ธโ‰ค๎€ท๐‘“,๐‘ขโ„Ž๎€ธ.(2.19)

Using Theorem 2.3, we deduce that12๎€œฮฉ๐‘‘๐‘ข2โ„Ž๎€ท๐‘ข๐‘‘๐‘ก๐‘‘๐‘ฅ+๐‘Žโ„Ž,๐‘ขโ„Ž๎€ธโ‰ฅ12๐‘‘๎€œ๐‘‘๐‘กฮฉ๐‘ข2โ„Žโ€–โ€–๐‘ข๐‘‘๐‘ฅ+๐›ผโ„Žโ€–โ€–21โ€–โ€–๐‘ขโˆ’๐œ†โ„Žโ€–โ€–22=12๐‘‘๎€œ๐‘‘๐‘กฮฉ๐‘ข2โ„Žโ€–โ€–๐‘ข๐‘‘๐‘ฅ+๐›ผโ„Žโ€–โ€–22โ€–โ€–+๐›ผโˆ‡๐‘ขโ„Žโ€–โ€–22โ€–โ€–๐‘ขโˆ’๐œ†โ„Žโ€–โ€–22=12๎‚€๐‘‘๐ธ๐‘‘๐‘กโ„Ž(๐‘ก)+2(๐›ผโˆ’๐œ†)๐ธโ„Žโ€–โ€–(๐‘ก)+2๐›ผโˆ‡๐‘ขโ„Žโ€–โ€–22๎‚โ‰ฅ1๐‘‘๐‘ฅ2๎‚ต๐‘‘๐ธ๐‘‘๐‘กโ„Ž(๐‘ก)+2(๐›ผโˆ’๐œ†)๐ธโ„Ž(๐‘ก)+2๐›ผ๐ถ2โˆ—๎€œฮฉ๐‘ข2โ„Ž๎‚ถ.๐‘‘๐‘ฅ(2.20)

Thus, we have๐‘‘๎€œ๐‘‘๐‘กฮฉ๐‘ข2โ„Žโ€–โ€–๐‘ข๐‘‘๐‘ฅ+๐›ผโ„Žโ€–โ€–21โ€–โ€–๐‘ขโˆ’๐œ†โ„Žโ€–โ€–22โ‰ฅ๐‘‘๎€ท๐ธ๐‘‘๐‘กโ„Ž๎€ธ๎‚ต๐›ผ(๐‘ก)+2๐›ผโˆ’๐œ†+๐ถ2โˆ—๎‚ถ๐ธโ„Ž(๐‘ก).(2.21)

Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find๎€ท๐‘“,๐‘ขโ„Ž๎€ธ=๎€œฮฉ๐‘“(๐‘ฅ,๐‘ก)๐‘ขโ„Ž(๐‘ฅ,๐‘ก)๐‘‘๐‘ฅโ‰คโ€–๐‘“โ€–2โ€–โ€–๐‘ขโ„Žโ€–โ€–2.(2.22)

So that ๐‘‘๐ธ๐‘‘๐‘กโ„Ž๎‚ต๐›ผ(๐‘ก)+2๐›ผโˆ’๐œ†+๐ถ2โˆ—๎‚ถ๐ธโ„Ž(๐‘ก)โ‰ค2โ€–๐‘“โ€–2โ€–โ€–๐‘ขโ„Žโ€–โ€–2.(2.23)

Using Youngโ€™s inequality๐‘Ž๐‘โ‰ค๐œ€๐‘Ž2+1๐‘4๐œ€2,โˆ€๐‘Ž,๐‘โˆˆโ„,โˆ€๐œ€>0.(2.24)

Thus, we obtain๐‘‘๐ธ๐‘‘๐‘กโ„Ž๎‚ต๐›ผ(๐‘ก)+2๐›ผโˆ’๐œ†+๐ถ2โˆ—๎‚ถ๐ธโ„Ž(๐‘ก)โ‰ค2๐œ€๐ธโ„Ž1(๐‘ก)+2๐œ€โ€–๐‘“โ€–22,(2.25)

taking ๐œ‚=๐›ผโˆ’๐œ†+๐›ผ/๐ถ2โˆ—, thus we have ๐‘‘๐ธ๐‘‘๐‘กโ„Ž(๐‘ก)+2(๐œ‚โˆ’๐œ€)๐ธโ„Ž1(๐‘ก)โ‰ค2๐œ€โ€–๐‘“โ€–22.(2.26)

Or, equivalently ๎€ท๐‘’2(๐œ‚โˆ’๐œ€)๐‘ก๐ธโ„Ž(๎€ธ๐‘ก)๎…žโ‰ค1๐‘’2๐œ€2(๐œ‚โˆ’๐œ€)๐‘ก๎€œฮฉ(๐‘“(๐‘ฅ,๐‘ก))2๐‘‘๐‘ฅ.(2.27)

Integrating the last inequality from 0 to ๐‘ก, we get ๐ธโ„Ž(๐‘ก)โ‰ค๐‘’โˆ’2(๐œ‚โˆ’๐œ€)๐‘ก๐ธโ„Ž(10)+๎€œ2๐œ€๐‘ก0๎‚ธ๐‘’2(๐œ‚โˆ’๐œ€)(๐‘ โˆ’๐‘ก)๎‚ต๎€œฮฉ(๐‘“(๐‘ฅ,๐‘ ))2๐‘‘๐‘ฅ๎‚ถ๎‚น๐‘‘๐‘ .(2.28)

Remark 2.4. In particular, when ๐‘“=0 and choosing ๐œ€<๐œ‚, then (2.28) shows that the energy ๐ธ(๐‘ก) decreasing exponentially fast in time.

3. The ๐œƒ-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 ๎‚ต๐œ•๐‘ขโ„Ž๐œ•๐‘ก,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๎‚ถ๎€ท๐‘ข+๐‘Žโ„Ž,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๎€ธโ‰ฅ๎€ท๐‘“,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๎€ธ,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž.(3.1)

Now, we apply the ๐œƒ-scheme on the semidiscrete problem (3.1); for any ๐œƒโˆˆ[0,1] and ๐‘˜=1,โ€ฆ,๐‘›, we have ๎€ท๐‘ข๐‘˜โ„Žโˆ’๐‘ขโ„Ž๐‘˜โˆ’1,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ๎€ท๐‘ข+ฮ”๐‘กโ‹…๐‘Žโ„Ž๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ๎€ท๐‘“โ‰ฅฮ”๐‘กโ‹…๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž,(3.2) where ๐‘ขโ„Ž๐œƒ,๐‘˜=๐œƒ๐‘ข๐‘˜โ„Ž+(1โˆ’๐œƒ)๐‘ขโ„Ž๐‘˜โˆ’1,๐‘“๐œƒ,๐‘˜๎€ท๐‘ก=๐œƒ๐‘“๐‘˜๎€ธ๎€ท๐‘ก+(1โˆ’๐œƒ)๐‘“๐‘˜โˆ’1๎€ธ.(3.3)

It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear form ๐‘Ž(โ‹…,โ‹…). We recall that ๐‘Š is compactly embedded in ๐ฟ2(ฮฉ), since ฮฉ is bounded. Thus, there exists a nondecreasing sequence of eigenvalues ๐›ฟโ‰ค๐œ†1โ‰ค๐œ†2โ‰คโ‹ฏ for the bilinear form ๐‘Ž(โ‹…,โ‹…) satisfying ๐œ”๐‘—โˆˆ๐ฟ2,๐œ”๐‘—๎€ท๐œ”โ‰ 0:๐‘Ž๐‘—,๐‘ฃโ„Ž๎€ธ=๐œ†๐‘—๎€ท๐œ”๐‘—,๐‘ฃโ„Ž๎€ธ,โˆ€๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž.(3.4) The corresponding eigenfunctions {๐œ”๐‘—} form a complete orthonormal basis in ๐ฟ2(ฮฉ). In analogous way, when considering the finite dimensional problem in ๐‘Šโ„Ž, we find a sequence of eigenvalues ๐›ฟโ‰ค๐œ†1โ„Žโ‰ค๐œ†2โ„Žโ‰คโ‹ฏโ‰ค๐œ†๐‘š(โ„Ž) and ๐ฟ2-orthonormal basis of eigenvectorss ๐œ”๐‘–โ„Žโˆˆ๐‘Šโ„Ž,๐‘–=1,2,โ€ฆ,๐‘š(โ„Ž). Any function ๐‘ฃโ„Ž in ๐‘‰โ„Ž can thus be expanded with respect to the system ๐œ”๐‘–โ„Ž as ๐‘ฃโ„Ž=๐‘š(โ„Ž)๎“๐‘–=1๎€ท๐‘ฃโ„Ž,๐œ”๐‘–โ„Ž๎€ธ๐œ”๐‘–โ„Ž,(3.5) in particular, we have ๐‘ข๐‘˜โ„Ž=๐‘š(โ„Ž)๎“๐‘–=1๐‘ข๐‘˜๐‘–๐œ”๐‘–โ„Ž,๐‘ข๐‘˜๐‘–=๎€ท๐‘ข๐‘˜โ„Ž,๐œ”๐‘–โ„Ž๎€ธ.(3.6) Moreover, let ๐‘“๐‘˜โ„Ž be the ๐ฟ2-orthogonal projection of ๐‘“๐œƒ,๐‘˜ into ๐‘Šโ„Ž, that is, ๐‘“๐‘˜โ„Žโˆˆ๐‘Šโ„Ž and ๎€ท๐‘“๐‘˜โ„Ž,๐‘ฃโ„Ž๎€ธ=๎€ท๐‘“๐œƒ,๐‘˜,๐‘ฃโ„Ž๎€ธ,(3.7) and set ๐‘“๐‘˜โ„Ž=๐‘š(โ„Ž)๎“๐‘–=1๐‘“๐‘˜๐‘–๐œ”๐‘–โ„Ž,๐‘“๐‘˜๐‘–=๎€ท๐‘“๐‘˜โ„Ž,๐œ”๐‘–โ„Ž๎€ธ.(3.8) We are now in a position to prove the stability for ๐œƒโˆˆ[0,1/2[

Choosing in (3.1) ๐‘ฃโ„Ž=0, thus we have 1๎€ท๐‘ขฮ”๐‘ก๐‘˜โ„Žโˆ’๐‘ขโ„Ž๐‘˜โˆ’1,๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ๎€ท๐‘ข+๐‘Žโ„Ž๐œƒ,๐‘˜,๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธโ‰ค๎€ท๐‘“๐œƒ,๐‘˜,๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ,๐‘ขโ„Ž๐œƒ,๐‘˜โˆˆ๐‘‰โ„Ž.(3.9)

The inequalities (3.2) is equivalent to 1๎€ท๐‘ขฮ”๐‘ก๐‘˜๐‘–โˆ’๐‘ข๐‘–๐‘˜โˆ’1๎€ธ+๐œ†๐‘–โ„Ž๎€ท๐œƒ๐‘ข๐‘˜๐‘–+(1โˆ’๐œƒ)๐‘ข๐‘–๐‘˜โˆ’1๎€ธโ‰ค๐‘“๐‘˜๐‘–.(3.10)

Since ๐œ”๐‘–โ„Ž are the eigenfunctions means ๐‘Ž๎€ท๐œ”๐‘–โ„Ž,๐œ”๐‘–โ„Ž๎€ธ=๐œ†๐‘–โ„Ž๎€ท๐œ”๐‘–โ„Ž,๐œ”๐‘–โ„Ž๎€ธ=๐œ†๐‘–โ„Žโ‹…๐›ฟ๐‘–๐‘–=๐œ†๐‘–โ„Ž,(3.11) for each ๐‘˜=0,โ€ฆ,๐‘š(โ„Ž)โˆ’1, we can rewrite (3.9) as ๐‘ข๐‘˜๐‘–โ‰ค1โˆ’(1โˆ’๐œƒ)โ‹…ฮ”๐‘กโ‹…๐œ†๐‘–โ„Ž1+๐œƒฮ”๐‘กโ‹…๐œ†๐‘–โ„Ž๐‘ข๐‘–๐‘˜โˆ’1+ฮ”๐‘ก1+๐œƒฮ”๐‘กโ‹…๐œ†๐‘–โ„Ž๐‘“๐‘˜๐‘–,(3.12) this inequality system stable if and only if ||||1โˆ’(1โˆ’๐œƒ)โ‹…ฮ”๐‘กโ‹…๐œ†๐‘–โ„Ž1+๐œƒฮ”๐‘กโ‹…๐œ†๐‘–โ„Ž||||<1,(3.13) that is to say 22๐œƒโˆ’1>โˆ’๐œ†๐‘–โ„Žโ‹…ฮ”๐‘ก(3.14) means 2ฮ”๐‘ก<(1โˆ’2๐œƒ)๐œ†๐‘–โ„Ž.(3.15)

So that this relation satisfied for all the eigenvalues ๐œ†๐‘–โ„Ž of bilinear form ๐‘Ž(โ‹…,โ‹…), we have to choose their highest value, and we take it for ๐œ†๐‘šโ„Ž=๐œŒ(๐ด) (rayon spectral)

We deduce that if ๐œƒโ‰ฅ1/2 the ๐œƒ-scheme way is stable unconditionally (i.e., stable for all ฮ”t). However, if 0โ‰ค๐œƒ<1/2 the ๐œƒ-scheme is stable unless 2ฮ”๐‘ก<(1โˆ’2๐œƒ)๐œŒ(๐ด).(3.16)

We can prove that there exist two positive constants ๐‘1,๐‘2 such that ๐‘1โ„Ž2โ‰ค๐œ†๐‘šโ„Ž=๐‘2โ„Žโˆ’2,(3.17) thus the method of ๐œƒ-scheme is stable if and only if ฮ”๐‘ก<2๐ถโ„Ž(1โˆ’2๐œƒ)2.(3.18)

Notice that this condition is always satisfied if 0โ‰ค๐œƒ<1/2. Hence, taking the absolute value of (3.12), we have ||๐‘ข๐‘š๐‘–||<||๐‘ข0๐‘–||+ฮ”๐‘ก1+๐œƒฮ”๐‘กโ‹…๐œ†๐‘–โ„Ž๐‘šโˆ’1๎“๐‘–=1๐‘“๐‘˜๐‘–,(3.19) also we deduce that โ€–โ€–๐‘ข๐‘š๐‘–โ€–โ€–โˆž<โ€–โ€–๐‘ข0๐‘–โ€–โ€–โˆž+โ€–โ€–โ€–ฮ”๐‘ก1+๐œƒฮ”๐‘กโ‹…๐œ†๐‘–โ„Žโ€–โ€–โ€–โˆž๐‘šโˆ’1๎“๐‘–=1โ€–โ€–๐‘“๐‘˜๐‘–โ€–โ€–โˆž.(3.20)

Remark 3.1 (cf. [4]). We assume that the coerciveness assumption (Theorem 2.3) is satisfied with ๐œ†=0, and for each ๐‘˜=1,โ€ฆ,๐‘›, we find โ€–โ€–๐‘ข๐‘˜โ„Žโ€–โ€–22+2ฮ”๐‘ก๐‘›๎“๐‘˜=1๐‘Ž๎€ท๐‘ขโ„Ž๐œƒ,๐‘˜,๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ๎ƒฉโ‰ค๐ถ(๐‘›)๐‘›๎“๐‘˜=1โ€–โ€–๐‘“ฮ”๐‘ก๐œƒ,๐‘˜โ€–โ€–22๎ƒช.(3.21)

4. Asymptotic Behavior of ๐œƒ-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 ๐ถ independent of โ„Ž such that โ€–โ€–๐‘ขโˆžโˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๐ถโ„Ž2||||logโ„Ž2,(4.1) where ๐‘ขโˆž and ๐‘ขโˆžโ„Ž are, respectively, stationery solutions to the following continue and discrete inequalities: ๐‘(๐‘ขโˆž,๐‘ฃโˆ’๐‘ขโˆž)โ‰ฅ(๐‘“+๐œ†๐‘ขโˆž,๐‘ฃโˆ’๐‘ขโˆž),๐‘ฃโˆˆ๐ป10๐‘๎€ท๐‘ข(ฮฉ),(4.2)โˆžโ„Ž,๐‘ฃโ„Žโˆ’๐‘ขโˆžโ„Ž๎€ธโ‰ฅ๎€ท๐‘“+๐œ†๐‘ขโˆž,๐‘ฃโ„Žโˆ’๐‘ขโˆžโ„Ž๎€ธ,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž,(4.3) such that ๐‘(โ‹…,โ‹…)=๐‘Ž(โ‹…,โ‹…)+๐œ†(โ‹…,โ‹…),(4.4)where ๐œ† is a positive constant arbitrary.We have ๐‘ขโ„Ž๐œƒ,๐‘˜=๐œƒ๐‘ข๐‘˜โ„Ž+(1โˆ’๐œƒ)๐‘ขโ„Ž๐‘˜โˆ’1โ‰ค๐œƒ๐‘Ÿโ„Ž๐œ“+(1โˆ’๐œƒ)๐‘Ÿโ„Ž๐œ“=๐‘Ÿโ„Ž๐œ“.Thus, we can rewrite (3.1) as, for ๐‘ขโ„Ž๐œƒ,๐‘˜โˆˆ๐‘‰โ„Ž๎ƒฉ๐‘ขโ„Ž๐œƒ,๐‘˜๐œƒฮ”๐‘ก,๐‘ฃโ„Žโˆ’ฬƒ๐‘ข๐‘˜โ„Ž๎ƒช๎€ท๐‘ข+๐‘Žโ„Ž๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธโ‰ฅ๎ƒฉ๐‘“๐œƒ,๐‘˜+๐‘ขโ„Ž๐‘˜โˆ’1๐œƒฮ”๐‘ก,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎ƒช,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž.(4.5) Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities: ๐‘๎€ท๐‘ขโ„Ž๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’ฬƒ๐‘ข๐‘˜โ„Ž๎€ธโ‰ฅ๎€ท๐‘“๐œƒ,๐‘˜+๐œ‡๐‘ขโ„Ž๐‘˜โˆ’1,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž,(4.6) such that ๐‘๎€ท๐‘ขโ„Ž๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ๎€ท๐‘ข=๐œ‡โ„Ž๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ๎€ท๐‘ข+๐‘Žโ„Ž๐œƒ,๐‘˜,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,๐‘˜๎€ธ,๐‘ฃโ„Ž,๐‘ขโ„Ž๐œƒ,๐‘˜โˆˆ๐‘‰โ„Ž,1๐œ‡==๐‘‡๐œƒฮ”๐‘ก,๐œƒ๐‘˜(4.7) where ๐‘ขโ„Ž๐œƒ,1 is the solution to the following discrete inequality: ๐‘Ž๎€ท๐‘ขโ„Ž๐œƒ,1,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,1๎€ธ=๎€ท๐‘”๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฃโ„Žโˆ’๐‘ขโ„Ž๐œƒ,1๎€ธ,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž,(4.8) where ๐‘”(๐‘ก๐‘˜) is a regular function given.

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

We consider the mapping ๐‘‡โ„Žโˆถ๐ฟโˆž+(ฮฉ)โŸถ๐‘‰โ„Ž๐‘คโŸถ๐‘‡โ„Ž(๐‘ค)=๐œ‰โ„Ž,(4.9) where ๐œ‰โ„Ž is the unique solution of the following P.V.I: find ๐œ‰โ„Žโˆˆ๐‘‰โ„Ž๐‘๎€ท๐œ‰โ„Ž,๐‘ฃโ„Žโˆ’๐œ‰โ„Ž๎€ธโ‰ฅ๎€ท๐‘“๐œƒ,๐‘˜+๐œ‡๐‘ค,๐‘ฃโ„Žโˆ’๐œ‰โ„Ž๎€ธ,๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž.(4.10)

Proposition 4.2. Under the previous hypotheses and notations, if one sets ๐œƒโ‰ฅ1/2, the mapping ๐‘‡โ„Ž is a contraction in ๐ฟโˆž(ฮฉ) with rate of contraction 1/(1+๐›ฝโ‹…๐œƒโ‹…ฮ”๐‘ก). Therefore, ๐‘‡โ„Ž admits a unique fixed point which coincides with the solution of P.V. I (4.7).

Proof. For ๐‘ค, ๎‚๐‘ค in ๐ฟโˆž(ฮฉ), we consider ๐œ‰โ„Ž=๐‘‡โ„Ž(๐‘ค)=๐œ•(๐‘“๐œƒ,๐‘˜+๐œ‡๐‘ค,๐‘Ÿโ„Ž๐œ“) and ฬƒ๐œ‰โ„Ž=๐‘‡โ„Ž(๎‚๐‘ค)=๐œ•(๐‘“๐œƒ,๐‘˜๎‚+๐œ‡๐‘ค,๐‘Ÿโ„Ž๎‚๐œ“) solution to quasivariational inequalities (4.7) with right-hand side ๐น๐œƒ,๐‘˜=๐‘“๐œƒ,๐‘˜+๐œ‡๐‘คโ„Ž,๎‚๐น๐œƒ,๐‘˜=๐‘“๐œƒ,๐‘˜๎‚๐‘ค+๐œ‡โ„Ž.
Now, setting 1๐œ™=โ€–โ€–๐น๐›ฝ+๐œ‡๐œƒ,๐‘˜โˆ’๎‚๐น๐œƒ,๐‘˜โ€–โ€–โˆž,(4.11) then for ๐œ‰โ„Ž+๐œ™ is solution of ๐‘๎€ท๐œ‰โ„Ž๎€ท๐‘ฃ+๐œ™,โ„Ž๎€ธโˆ’๎€ท๐œ‰+๐œ™โ„Žโ‰ฅ๎€ท๐น+๐œ™๎€ธ๎€ธ๐œƒ,๐‘˜+๐‘Ž0๎€ท๐‘ฃ๐œ™,โ„Ž๎€ธโˆ’๎€ท๐œ‰+๐œ™โ„Ž,+๐œ™๎€ธ๎€ธ๐œ‰+๐œ™โ‰ค๐‘Ÿโ„Ž๐œ“+๐œ™,๐‘ฃโ„Ž+๐œ™โ‰ค๐‘Ÿโ„Ž๐œ“+๐œ™,โˆ€๐‘ฃโ„Žโˆˆ๐‘‰โ„Ž.(4.12) Also, we have ๐น๐œƒ,๐‘˜โ‰ค๎‚๐น๐œƒ,๐‘˜+โ€–โ€–๐น๐œƒ,๐‘˜โˆ’๎‚๐น๐œƒ,๐‘˜โ€–โ€–โˆžโ‰ค๎‚๐น๐œƒ,๐‘˜+๐‘Ž0โ€–โ€–๐น๐›ฝ+๐œ‡๐œƒ,๐‘˜โˆ’๎‚๐น๐œƒ,๐‘˜โ€–โ€–โˆžโ‰ค๎‚๐น๐œƒ,๐‘˜+๐‘Ž0๐œ™,(4.13) thus ๐œ•โ„Ž๎€ท๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๎€ธ๐œ“+๐œ™โ‰ค๐œ•โ„Ž๎‚€๎‚๐น๐œƒ,๐‘˜+๐‘Ž0(๐‘ฅ)๐œ™,๐‘Ÿโ„Ž๎‚๎‚๐œ“+๐œ™โ‰ค๐œ•โ„Ž๎‚€๎‚๐น๐œƒ,๐‘˜๎‚,๐‘Ÿ๎‚๐œ“+๐œ™,(4.14) hence ๐œ‰โ„Žโ‰คฬƒ๐œ‰โ„Ž+๐œ™.(4.15) Similarly, interchanging the roles of ๐‘ค and ๎‚๐‘ค, we also get ฬƒ๐œ‰โ„Žโ‰ค๐œ‰โ„Ž+๐œ™.(4.16) Finally, this yields โ€–โ€–๐œ•โ„Ž๎€ท๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๐œ“๎€ธโˆ’๐œ•โ„Ž๎‚€๎‚๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๎‚โ€–โ€–๎‚๐œ“โˆžโ‰ค1โ€–โ€–๐น๐›ฝ+๐œ‡๐œƒ,๐‘˜โˆ’๎‚๐น๐œƒ,๐‘˜โ€–โ€–โˆžโ‰ค1โ€–โ€–๐‘“๐›ฝ+๐œ‡๐œƒ,๐‘˜+๐œ‡๐‘คโˆ’๐‘“๐œƒ,๐‘˜๎‚๐‘คโ€–โ€–โˆ’๐œ‡โˆžโ€–โ€–๎‚๐‘คโ€–โ€–๐‘คโˆ’โˆžโ‰ค1โ€–โ€–๎‚๐‘คโ€–โ€–1+๐›ฝ๐œƒฮ”๐‘ก๐‘คโˆ’โˆž,(4.17) which completes the proof.

Remark 4.3. If we set 0โ‰ค๐œƒ<1/2, the mapping ๐‘‡โ„Ž is a contraction in ๐ฟโˆž(ฮฉ) with rate of contraction 2/(2+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐œŒ(๐ด)), where ๐œŒ(๐ด) is a spectral radius of operator ๐ด.

Proof. Under condition of stability, we have shown the ๐œƒ-scheme is stable if and only if ฮ”๐‘ก<(2๐ถ/(1โˆ’2๐œƒ))โ„Ž2.
Thus it can be easily show that โ€–โ€–๐œ•โ„Ž๎€ท๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๐œ“๎€ธโˆ’๐œ•โ„Ž๎‚€๎‚๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๎‚โ€–โ€–๎‚๐œ“โˆžโ‰ค1โ€–โ€–๎‚๐‘คโ€–โ€–1+๐›ฝ๐œƒฮ”๐‘ก๐‘คโˆ’โˆžโ‰ค2โ€–โ€–๎‚๐‘คโ€–โ€–2+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐œŒ(๐ด)๐‘คโˆ’โˆžโ‰ค1๎€ท1+๐›ฝ๐œƒ(1โˆ’2๐œƒ)/2๐ถโ„Ž2๎€ธโ€–โ€–๎‚๐‘คโ€–โ€–๐‘คโˆ’โˆž=2๐ถโ„Ž22๐ถโ„Ž2โ€–โ€–๎‚๐‘คโ€–โ€–+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐‘คโˆ’โˆž,(4.18) also it can be found that โ€–โ€–๐œ•โ„Ž๎€ท๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๐œ“๎€ธโˆ’๐œ•โ„Ž๎‚€๎‚๐น๐œƒ,๐‘˜,๐‘Ÿโ„Ž๎‚โ€–โ€–๎‚๐œ“โˆžโ‰ค1๎€ท(1+๐›ฝ๐œƒ1โˆ’2๐œƒ)/2๐ถโ„Ž2๎€ธโ€–โ€–๎‚๐‘คโ€–โ€–๐‘คโˆ’โˆž=2๐ถโ„Ž22๐ถโ„Ž2โ€–โ€–๎‚๐‘คโ€–โ€–+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐‘คโˆ’โˆž,(4.19) thus the mapping ๐‘‡โ„Ž is a contraction in ๐ฟโˆž(ฮฉ) with rate of contraction (2๐ถโ„Ž2)/(2๐ถโ„Ž2+๐›ฝ๐œƒ(1โˆ’2๐œƒ)). Therefore, ๐‘‡โ„Ž admits a unique fixed point which coincides with the solution of P.V.I (4.7) โ€–โ€–๐‘‡โ„Ž(๐‘ค)โˆ’Tโ„Ž๎€ท๎‚๐‘ค๎€ธโ€–โ€–โˆžโ‰ค1โ€–โ€–๐น๐›ฝ+๐œ‡๐œƒ,๐‘˜โˆ’๎‚๐น๐œƒ,๐‘˜โ€–โ€–โˆž=1โ€–โ€–๐‘“๐›ฝ+๐œ‡๐œƒ,๐‘˜+๐œ‡๐‘คโˆ’๐‘“๐œƒ,๐‘˜๎‚๐‘คโ€–โ€–โˆ’๐œ‡โˆžโ‰ค๐œ‡โ€–โ€–๎‚๐‘คโ€–โ€–๐›ฝ+๐œ‡๐‘คโˆ’โˆžโ‰ค1โ€–โ€–๎‚๐‘คโ€–โ€–1+๐›ฝ๐œƒฮ”๐‘ก๐‘คโˆ’โˆž.(4.20) This completes the proof.

4.2. Discrete Algorithm

Starting from ๐‘ข0โ„Ž=๐‘ข0โ„Ž (initial data) and the ๐‘ขโ„Ž๐œƒ,1 solution of problem (4.7), we introduce the following discrete algorithm: ๐‘ขโ„Ž๐œƒ,๐‘˜=๐‘‡โ„Ž๐‘ขโ„Ž๐‘˜โˆ’1,๐‘˜=1,โ€ฆ,๐‘›,(4.21) where ๐‘ขโ„Ž๐œƒ,๐‘˜ is the solution of the problem (4.7).

Remark 4.4. If we choose ๐œƒ=1 in (4.21), we get Bensoussanโ€™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 ๐œƒโ‰ฅ1/2โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘˜โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต1๎‚ถ1+๐›ฝ๐œƒฮ”๐‘ก๐‘˜โ€–โ€–๐‘ขโˆžโ„Žโˆ’๐‘ขโ„Ž0โ€–โ€–โˆž,(4.22) and one has for โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘˜โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต2๐ถโ„Ž22๐ถโ„Ž2๎‚ถ+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐‘˜โ€–โ€–๐‘ขโˆžโ„Žโˆ’๐‘ขโ„Ž0โ€–โ€–โˆž1for0โ‰ค๐œƒ<2.(4.23)

Proof. we set a first case ๐œƒโ‰ฅ1/2, and we have ๐‘ขโˆžโ„Ž=๐‘‡โ„Ž๐‘ขโˆžโ„Ž,โ€–โ€–๐‘ขโ„Ž๐œƒ,1โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž=โ€–โ€–๐‘‡โ„Ž๐‘ข0โ„Žโˆ’๐‘‡โ„Ž๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต1๎‚ถโ€–โ€–๐‘ข1+๐›ฝ๐œƒฮ”๐‘ก0โ„Žโˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž.(4.24) for ๐‘˜โ‰ฅ2, we use the Bensoussan-Lions' algorithm (๐‘ข๐‘˜โ„Ž=๐‘‡โ„Ž๐‘ขโ„Ž๐‘˜โˆ’1,๐‘˜=1,โ€ฆ,๐‘›) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘˜โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต1๎‚ถ1+๐›ฝ๐œƒฮ”๐‘ก๐‘˜โ€–โ€–๐‘ข0โ„Žโˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž,(4.25) so โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘˜+1โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž=โ€–โ€–๐‘‡โ„Ž๐‘ข๐‘˜โ„Žโˆ’๐‘‡โ„Ž๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต1๎‚ถโ€–โ€–๐‘ข1+๐›ฝ๐œƒฮ”๐‘ก๐‘˜โ„Žโˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž,(4.26) thus โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘˜+1โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต1๎‚ถ1+๐›ฝ๐œƒฮ”๐‘ก๐‘˜+1โ€–โ€–๐‘ข0โ„Žโˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž,(4.27) for a second case 0โ‰ค๐œƒ<1/2, it can be easily shown that โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘˜โˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆžโ‰ค๎‚ต2๐ถโ„Ž22๐ถโ„Ž2๎‚ถ+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐‘˜โ€–โ€–๐‘ข0โ„Žโˆ’๐‘ขโˆžโ„Žโ€–โ€–โˆž.(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 ๐ฟโˆž-norm for parabolic variational inequalities

Now, we evaluate the variation in ๐ฟโˆž between ๐‘ข๐œƒโ„Ž(๐‘‡,๐‘ฅ), the discrete solution calculated at the moment ๐‘‡=๐‘›ฮ”๐‘ก and ๐‘ขโˆž, 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 ๐œƒโ‰ฅ1/2, โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘›โˆ’๐‘ขโˆžโ€–โ€–โˆž๎‚ธโ„Žโ‰ค๐ถ2||||logโ„Ž2+๎‚ต1๎‚ถ1+๐›ฝ๐œƒฮ”๐‘ก๐‘›๎‚น,(4.29) and for the second case 0โ‰ค๐œƒ<1/2, โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘›โˆ’๐‘ขโˆžโ€–โ€–โˆž๎‚ธโ„Žโ‰ค๐ถ2||||logโ„Ž2+๎‚ต2๐ถโ„Ž22๐ถโ„Ž2๎‚ถ+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐‘›๎‚น,(4.30) where ๐ถ is a constant independent of โ„Ž and ๐‘˜.

Proof. We have ๐‘ขโ„Ž๐œƒ,๐‘˜(๐‘ฅ)=๐‘ขโ„Ž][(๐‘ก,๐‘ฅ)for๐‘กโˆˆ(๐‘˜โˆ’1)ฮ”๐‘ก;๐‘˜ฮ”๐‘ก,(4.31) thus ๐‘ขโ„Ž๐œƒ,๐‘›(๐‘ฅ)=๐‘ขโ„Ž(๐‘‡,๐‘ฅ),(4.32) then โ€–โ€–๐‘ข๐œƒโ„Ž(๐‘‡,๐‘ฅ)โˆ’๐‘ขโˆžโ€–โ€–๐ฟโˆž(ฮฉ)=โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘›โˆ’๐‘ขโˆžโ€–โ€–๐ฟโˆž(ฮฉ)โ‰คโ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘›โˆ’๐‘ขโˆžโ„Žโ€–โ€–๐ฟโˆž(ฮฉ)+โ€–โ€–๐‘ขโˆžโ„Žโˆ’๐‘ขโˆžโ€–โ€–๐ฟโˆž(ฮฉ).(4.33) Using, Theorem 4.1 and Proposition 4.5, we have for ๐œƒโ‰ฅ1/2, โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘›โˆ’๐‘ขโˆžโ€–โ€–โˆž๎‚ธโ„Žโ‰ค๐ถ2||||logโ„Ž2+๎‚ต1๎‚ถ1+๐›ฝ๐œƒฮ”๐‘ก๐‘›๎‚น,(4.34) and for 0โ‰ค๐œƒ<1/2, we have โ€–โ€–๐‘ขโ„Ž๐œƒ,๐‘›โˆ’๐‘ขโˆžโ€–โ€–โˆž๎‚ธโ„Žโ‰ค๐ถ2||||logโ„Ž2+๎‚ต2๐ถโ„Ž22๐ถโ„Ž2๎‚ถ+๐›ฝ๐œƒ(1โˆ’2๐œƒ)๐‘›๎‚น.(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.

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Copyright ยฉ 2011 Salah Boulaaras and Mohamed Haiour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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