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

Authors: Salah Boulaaras; Mohamed Haiour
Journal: ISRN Mathematical Analysis
 (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.


---

## Body

## 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 theθ time scheme combined with a finite element spatial approximation for parabolic inequalities.Let us assume thatK is an implicit convex set defined as follows:
(1.1)K={v(t,x)∈L2(0,T,H01(Ω)),v(t,x)≤ψ(t,x),v(0,x)=v0inΩ},
with
(1.2)ψ∈L2(0,T,W2,∞(Ω)).We consider the following problem, findu∈K solution of
(1.3)∂u∂t+Au≤finΣ,u(t,x)=0inΓ,
where Σ is a set in ℝ×ℝN defined as Σ=Ω×[0,T] with T·<+∞, and Ω is convex domain in ℝN, 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
(1.4)Au=-∑ij=1N∂∂xiaij(x)∂u∂xj+∑Nj=1bj(x)∂u∂xj+a0(x)u,
and whose coefficients: ai,j(x),bj(x),a0(x)∈L∞(Ω)∩C2(Ω¯),x∈Ω¯,1≤i,j≤N are sufficiently smooth functions and satisfy the following conditions:
(1.5)aij(x)=aji(x);a0(x)≥β>0,βis a constant,(1.6)∑Nij=1aij(x)ξiξj≥γ|ξ|2;ξ∈RN,γ>0,x∈Ω¯,f is a regular functions satisfying
(1.7)f∈L2(0,T,L∞(Ω))∩C1(0,T,H-1(Ω)),f≥0.We specify the following notations:(1.8)‖⋅‖L2(Ω)=‖⋅‖2,‖⋅‖1=‖⋅‖H01(Ω),‖⋅‖L∞(Ω)=‖⋅‖∞.As we have said before, the aim of the present paper is to show that the asymptotic behavior can be properly approximated by aθ 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 H01 by a space discretization of finite dimensional Vh⊂H01. In the second step, we discretize the problem with respect to time using the θ-scheme. Therefore, we search a sequence of elements uhn∈Vh which approaches un(tn),tn=nΔt, with initial data uh0=u0h. Our approach stands on a discrete stability result and error estimate for parabolic variational inequalities.The paper is organized as follows. In Section2, 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:
(2.1)(∂u∂t,v-u)+a(u,v-u)≥(f,v-u),v∈K,
where a(·,·) is the bilinear form associated with operator A defined in (1.4). Namely,
(2.2)a(u,v)=∫Ω(∑Nij=1aij(x)∂u∂xi∂v∂xj+∑Nj=1bj(x)∂u∂xjv+a0(x)uv)dx,Theorem 2.1 (see [9]).
The problem (1.3) has an unique solution u∈K(u). Moreover, one has
(2.3)u∈L2(0,T;H01(Ω)),∂u∂t∈L2(0,T;H-1(Ω)).Lemma 2.2 (Sobolev-Poincare inequality).
LetΩ be a bounded overt in ℝN, with sufficiently smooth boundary Γ, then there exists a C* such that
(2.4)‖u‖2≤C*‖∇u‖2,v∈H01(Ω)∩C2(Ω¯),∇=∑Ni=1∂∂xi.


### 2.1. The Discrete Problem

Let us assume thatΩ can be decomposed into triangles and τh denotes the set of all the elements h>0, where h is the mesh size. We assume that the family τh is regular and quasi-uniform, and we consider the usual basis of affine functions φi, i={1,…,m(h)} defined by φi(Mj)=δij, where Mj is a vertex of the considered triangulation. We introduce the following discrete spaces Vh of finite element:
(2.5)Vh={vh∈L2(0,T,H01(Ω))∩C(0,T,H01(Ω¯)),suchthatvh|k∈P1,k∈τh,vh≤rhψ,vh(⋅,0)=vh0inΩ.vh∈L2(0,T,H01(Ω))∩C(0,T,H01(Ω¯))}.We considerrh to be the usual interpolation operator defined by
(2.6)v∈L2(0,T,H01(Ω))∩C(0,T,H01(Ω¯)),rhv=∑m(h)i=1v(Mi)φi(x).The Discrete Maximum Principle Assumption (see [10])
The matrix whose coefficientsa(φi,φj) 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 forma(·,·) defined as (2.2) is weakly coercive in Vh⊂H01(Ω). Then, there exists two constants α>0 and λ>0 such that
(2.7)a(uh,uh)+λ‖uh‖2≥α‖uh‖1,
where
(2.8)λ=(‖bj‖∞22γ+γ2+‖a0‖∞),α=γ2.Proof.
The bilinear forma(·,·) is defined by
(2.9)a(uh,uh)=∫Ω(∑Nij=1aij(x)∂uh∂xi∂uh∂xj+∑Nj=1bj(x)∂uh∂xjuh+a0(x)uh2)dx,
under assumption (1.6), we have
(2.10)∑Nij=1∫Ωaij(x)∂uh∂xi∂uh∂xj>γ∑i=1N(∫Ω∂uh∂xi)2=γ‖∇uh‖22,
and since
(2.11)|∑Nj=1∫Ωbj(x)∂uh∂xjuh|≤sup⁡j|bj|∑Nj=1∫Ω|∂uh∂xjuh|≤‖bj‖∞‖∇uh‖2‖uh‖2,
then we make use of the algebraic inequality
(2.12)ab≤12(a2+b2),∀a,b∈R,∀γ>0,
and choosing
(2.13)a=‖∇uh‖2⋅γ,b=‖bj‖∞γ‖uh‖2,
then we end up with
(2.14)|∑Nj=1∫Ωbj(x)∂uh∂xjuh|≥-(γ2‖∇uh‖22+‖bj‖∞⋅‖uh‖22γ),
so we get
(2.15)a(uh,uh)≥γ‖∇uh‖22-γ2‖∇uh‖22+‖bj‖∞2γ‖uh‖2-‖a0‖∞‖uh‖22.
It can easily verified that
(2.16)a(uh,uh)≥γ2(‖∇uh‖2+‖uh‖22)-(‖aj‖∞22γ+γ2+‖a0‖∞)‖uh‖22.
Consequently, we deduce from above that
(2.17)a(uh,uh)+λ‖uh‖22≥α‖uh‖12suchthatα=γ2,λ=(‖aj‖∞22γ+γ2+‖a0‖∞).We can identify the following result on the time energy behavior:(2.18)Eh(t)=∫Ωuh2dx.Settingv=0 on (2.1) and after discretization by the finite element in the Vh, we have the semidiscretization problem
(2.19)(∂uh∂t,uh)+a(uh,uh)=12∫Ω∂uh2∂tdx+a(uh,uh)≤(f,uh).Using Theorem2.3, we deduce that(2.20)12∫Ωduh2dtdx+a(uh,uh)≥12ddt∫Ωuh2dx+α‖uh‖12-λ‖uh‖22=12ddt∫Ωuh2dx+α‖uh‖22+α‖∇uh‖22-λ‖uh‖22=12(ddtEh(t)+2(α-λ)Eh(t)+2α‖∇uh‖22dx)≥12(ddtEh(t)+2(α-λ)Eh(t)+2αC*2∫Ωuh2dx).Thus, we have(2.21)ddt∫Ωuh2dx+α‖uh‖12-λ‖uh‖22≥ddt(Eh(t))+2(α-λ+αC*2)Eh(t).Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find(2.22)(f,uh)=∫Ωf(x,t)uh(x,t)dx≤‖f‖2‖uh‖2.So that(2.23)ddtEh(t)+2(α-λ+αC*2)Eh(t)≤2‖f‖2‖uh‖2.Using Young’s inequality(2.24)ab≤ɛa2+14ɛb2,∀a,b∈R,∀ɛ>0.Thus, we obtain(2.25)ddtEh(t)+2(α-λ+αC*2)Eh(t)≤2ɛEh(t)+12ɛ‖f‖22,takingη=α-λ+α/C*2, thus we have
(2.26)ddtEh(t)+2(η-ɛ)Eh(t)≤12ɛ‖f‖22.Or, equivalently(2.27)(e2(η-ɛ)tEh(t))′≤12ɛe2(η-ɛ)t∫Ω(f(x,t))2dx.Integrating the last inequality from 0 tot, we get
(2.28)Eh(t)≤e-2(η-ɛ)tEh(0)+12ɛ∫0t[e2(η-ɛ)(s-t)(∫Ω(f(x,s))2dx)]ds.Remark 2.4.
In particular, whenf=0 and choosing ε<η, then (2.28) shows that the energy E(t) decreasing exponentially fast in time.


## 2.1. The Discrete Problem

Let us assume thatΩ can be decomposed into triangles and τh denotes the set of all the elements h>0, where h is the mesh size. We assume that the family τh is regular and quasi-uniform, and we consider the usual basis of affine functions φi, i={1,…,m(h)} defined by φi(Mj)=δij, where Mj is a vertex of the considered triangulation. We introduce the following discrete spaces Vh of finite element:
(2.5)Vh={vh∈L2(0,T,H01(Ω))∩C(0,T,H01(Ω¯)),suchthatvh|k∈P1,k∈τh,vh≤rhψ,vh(⋅,0)=vh0inΩ.vh∈L2(0,T,H01(Ω))∩C(0,T,H01(Ω¯))}.We considerrh to be the usual interpolation operator defined by
(2.6)v∈L2(0,T,H01(Ω))∩C(0,T,H01(Ω¯)),rhv=∑m(h)i=1v(Mi)φi(x).The Discrete Maximum Principle Assumption (see [10])
The matrix whose coefficientsa(φi,φj) 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 forma(·,·) defined as (2.2) is weakly coercive in Vh⊂H01(Ω). Then, there exists two constants α>0 and λ>0 such that
(2.7)a(uh,uh)+λ‖uh‖2≥α‖uh‖1,
where
(2.8)λ=(‖bj‖∞22γ+γ2+‖a0‖∞),α=γ2.Proof.
The bilinear forma(·,·) is defined by
(2.9)a(uh,uh)=∫Ω(∑Nij=1aij(x)∂uh∂xi∂uh∂xj+∑Nj=1bj(x)∂uh∂xjuh+a0(x)uh2)dx,
under assumption (1.6), we have
(2.10)∑Nij=1∫Ωaij(x)∂uh∂xi∂uh∂xj>γ∑i=1N(∫Ω∂uh∂xi)2=γ‖∇uh‖22,
and since
(2.11)|∑Nj=1∫Ωbj(x)∂uh∂xjuh|≤sup⁡j|bj|∑Nj=1∫Ω|∂uh∂xjuh|≤‖bj‖∞‖∇uh‖2‖uh‖2,
then we make use of the algebraic inequality
(2.12)ab≤12(a2+b2),∀a,b∈R,∀γ>0,
and choosing
(2.13)a=‖∇uh‖2⋅γ,b=‖bj‖∞γ‖uh‖2,
then we end up with
(2.14)|∑Nj=1∫Ωbj(x)∂uh∂xjuh|≥-(γ2‖∇uh‖22+‖bj‖∞⋅‖uh‖22γ),
so we get
(2.15)a(uh,uh)≥γ‖∇uh‖22-γ2‖∇uh‖22+‖bj‖∞2γ‖uh‖2-‖a0‖∞‖uh‖22.
It can easily verified that
(2.16)a(uh,uh)≥γ2(‖∇uh‖2+‖uh‖22)-(‖aj‖∞22γ+γ2+‖a0‖∞)‖uh‖22.
Consequently, we deduce from above that
(2.17)a(uh,uh)+λ‖uh‖22≥α‖uh‖12suchthatα=γ2,λ=(‖aj‖∞22γ+γ2+‖a0‖∞).We can identify the following result on the time energy behavior:(2.18)Eh(t)=∫Ωuh2dx.Settingv=0 on (2.1) and after discretization by the finite element in the Vh, we have the semidiscretization problem
(2.19)(∂uh∂t,uh)+a(uh,uh)=12∫Ω∂uh2∂tdx+a(uh,uh)≤(f,uh).Using Theorem2.3, we deduce that(2.20)12∫Ωduh2dtdx+a(uh,uh)≥12ddt∫Ωuh2dx+α‖uh‖12-λ‖uh‖22=12ddt∫Ωuh2dx+α‖uh‖22+α‖∇uh‖22-λ‖uh‖22=12(ddtEh(t)+2(α-λ)Eh(t)+2α‖∇uh‖22dx)≥12(ddtEh(t)+2(α-λ)Eh(t)+2αC*2∫Ωuh2dx).Thus, we have(2.21)ddt∫Ωuh2dx+α‖uh‖12-λ‖uh‖22≥ddt(Eh(t))+2(α-λ+αC*2)Eh(t).Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find(2.22)(f,uh)=∫Ωf(x,t)uh(x,t)dx≤‖f‖2‖uh‖2.So that(2.23)ddtEh(t)+2(α-λ+αC*2)Eh(t)≤2‖f‖2‖uh‖2.Using Young’s inequality(2.24)ab≤ɛa2+14ɛb2,∀a,b∈R,∀ɛ>0.Thus, we obtain(2.25)ddtEh(t)+2(α-λ+αC*2)Eh(t)≤2ɛEh(t)+12ɛ‖f‖22,takingη=α-λ+α/C*2, thus we have
(2.26)ddtEh(t)+2(η-ɛ)Eh(t)≤12ɛ‖f‖22.Or, equivalently(2.27)(e2(η-ɛ)tEh(t))′≤12ɛe2(η-ɛ)t∫Ω(f(x,t))2dx.Integrating the last inequality from 0 tot, we get
(2.28)Eh(t)≤e-2(η-ɛ)tEh(0)+12ɛ∫0t[e2(η-ɛ)(s-t)(∫Ω(f(x,s))2dx)]ds.Remark 2.4.
In particular, whenf=0 and choosing ε<η, then (2.28) shows that the energy E(t) decreasing exponentially fast in time.


## 2.1.1. Priory Estimate

Theorem 2.3.
Let us assume that the discrete bilinear forma(·,·) defined as (2.2) is weakly coercive in Vh⊂H01(Ω). Then, there exists two constants α>0 and λ>0 such that
(2.7)a(uh,uh)+λ‖uh‖2≥α‖uh‖1,
where
(2.8)λ=(‖bj‖∞22γ+γ2+‖a0‖∞),α=γ2.Proof.
The bilinear forma(·,·) is defined by
(2.9)a(uh,uh)=∫Ω(∑Nij=1aij(x)∂uh∂xi∂uh∂xj+∑Nj=1bj(x)∂uh∂xjuh+a0(x)uh2)dx,
under assumption (1.6), we have
(2.10)∑Nij=1∫Ωaij(x)∂uh∂xi∂uh∂xj>γ∑i=1N(∫Ω∂uh∂xi)2=γ‖∇uh‖22,
and since
(2.11)|∑Nj=1∫Ωbj(x)∂uh∂xjuh|≤sup⁡j|bj|∑Nj=1∫Ω|∂uh∂xjuh|≤‖bj‖∞‖∇uh‖2‖uh‖2,
then we make use of the algebraic inequality
(2.12)ab≤12(a2+b2),∀a,b∈R,∀γ>0,
and choosing
(2.13)a=‖∇uh‖2⋅γ,b=‖bj‖∞γ‖uh‖2,
then we end up with
(2.14)|∑Nj=1∫Ωbj(x)∂uh∂xjuh|≥-(γ2‖∇uh‖22+‖bj‖∞⋅‖uh‖22γ),
so we get
(2.15)a(uh,uh)≥γ‖∇uh‖22-γ2‖∇uh‖22+‖bj‖∞2γ‖uh‖2-‖a0‖∞‖uh‖22.
It can easily verified that
(2.16)a(uh,uh)≥γ2(‖∇uh‖2+‖uh‖22)-(‖aj‖∞22γ+γ2+‖a0‖∞)‖uh‖22.
Consequently, we deduce from above that
(2.17)a(uh,uh)+λ‖uh‖22≥α‖uh‖12suchthatα=γ2,λ=(‖aj‖∞22γ+γ2+‖a0‖∞).We can identify the following result on the time energy behavior:(2.18)Eh(t)=∫Ωuh2dx.Settingv=0 on (2.1) and after discretization by the finite element in the Vh, we have the semidiscretization problem
(2.19)(∂uh∂t,uh)+a(uh,uh)=12∫Ω∂uh2∂tdx+a(uh,uh)≤(f,uh).Using Theorem2.3, we deduce that(2.20)12∫Ωduh2dtdx+a(uh,uh)≥12ddt∫Ωuh2dx+α‖uh‖12-λ‖uh‖22=12ddt∫Ωuh2dx+α‖uh‖22+α‖∇uh‖22-λ‖uh‖22=12(ddtEh(t)+2(α-λ)Eh(t)+2α‖∇uh‖22dx)≥12(ddtEh(t)+2(α-λ)Eh(t)+2αC*2∫Ωuh2dx).Thus, we have(2.21)ddt∫Ωuh2dx+α‖uh‖12-λ‖uh‖22≥ddt(Eh(t))+2(α-λ+αC*2)Eh(t).Applying the Cauchy-Schwartz inequality on the right-hand side of (2.1), we find(2.22)(f,uh)=∫Ωf(x,t)uh(x,t)dx≤‖f‖2‖uh‖2.So that(2.23)ddtEh(t)+2(α-λ+αC*2)Eh(t)≤2‖f‖2‖uh‖2.Using Young’s inequality(2.24)ab≤ɛa2+14ɛb2,∀a,b∈R,∀ɛ>0.Thus, we obtain(2.25)ddtEh(t)+2(α-λ+αC*2)Eh(t)≤2ɛEh(t)+12ɛ‖f‖22,takingη=α-λ+α/C*2, thus we have
(2.26)ddtEh(t)+2(η-ɛ)Eh(t)≤12ɛ‖f‖22.Or, equivalently(2.27)(e2(η-ɛ)tEh(t))′≤12ɛe2(η-ɛ)t∫Ω(f(x,t))2dx.Integrating the last inequality from 0 tot, we get
(2.28)Eh(t)≤e-2(η-ɛ)tEh(0)+12ɛ∫0t[e2(η-ɛ)(s-t)(∫Ω(f(x,s))2dx)]ds.Remark 2.4.
In particular, whenf=0 and choosing ε<η, then (2.28) shows that the energy E(t) decreasing exponentially fast in time.


## 3. Theθ-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
(3.1)(∂uh∂t,vh-uh)+a(uh,vh-uh)≥(f,vh-uh),vh∈Vh.Now, we apply theθ-scheme on the semidiscrete problem (3.1); for any θ∈[0,1] and k=1,…,n, we have
(3.2)(uhk-uhk-1,vh-uhθ,k)+Δt⋅a(uhθ,k,vh-uhθ,k)≥Δt⋅(fθ,k,vh-uhθ,k),vh∈Vh,
where
(3.3)uhθ,k=θuhk+(1-θ)uhk-1,fθ,k=θf(tk)+(1-θ)f(tk-1).It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear forma(·,·). 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
(3.4)ωj∈L2,ωj≠0:a(ωj,vh)=λj(ωj,vh),∀vh∈Vh.
The corresponding eigenfunctions {ωj} form a complete orthonormal basis in L2(Ω). In analogous way, when considering the finite dimensional problem in Wh, we find a sequence of eigenvalues δ≤λ1h≤λ2h≤⋯≤λm(h) and L2-orthonormal basis of eigenvectorss ωih∈Wh,i=1,2,…,m(h). Any function vh in Vh can thus be expanded with respect to the system ωih as
(3.5)vh=∑i=1m(h)(vh,ωih)ωih,
in particular, we have
(3.6)uhk=∑i=1m(h)uikωih,uik=(uhk,ωih).
Moreover, let fhk be the L2-orthogonal projection of fθ,k into Wh, that is, fhk∈Wh and
(3.7)(fhk,vh)=(fθ,k,vh),
and set
(3.8)fhk=∑i=1m(h)fikωih,fik=(fhk,ωih).
We are now in a position to prove the stability for θ∈[0,1/2[Choosing in (3.1) vh=0, thus we have
(3.9)1Δt(uhk-uhk-1,uhθ,k)+a(uhθ,k,uhθ,k)≤(fθ,k,uhθ,k),uhθ,k∈Vh.The inequalities (3.2) is equivalent to
(3.10)1Δt(uik-uik-1)+λih(θuik+(1-θ)uik-1)≤fik.Sinceωih are the eigenfunctions means
(3.11)a(ωih,ωih)=λih(ωih,ωih)=λih⋅δii=λih,
for each k=0,…,m(h)-1, we can rewrite (3.9) as
(3.12)uik≤1-(1-θ)⋅Δt⋅λih1+θΔt⋅λihuik-1+Δt1+θΔt⋅λihfik,
this inequality system stable if and only if
(3.13)|1-(1-θ)⋅Δt⋅λih1+θΔt⋅λih|<1,
that is to say
(3.14)2θ-1>-2λih⋅Δt
means
(3.15)Δt<2(1-2θ)λih.So that this relation satisfied for all the eigenvaluesλih of bilinear form a(·,·), we have to choose their highest value, and we take it for λmh=ρ(A) (rayon spectral)We deduce that ifθ≥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)Δt<2(1-2θ)ρ(A).We can prove that there exist two positive constantsc1,c2 such that
(3.17)c1h2≤λmh=c2h-2,
thus the method of θ-scheme is stable if and only if
(3.18)Δt<2C(1-2θ)h2.Notice that this condition is always satisfied if0≤θ<1/2. Hence, taking the absolute value of (3.12), we have
(3.19)|uim|<|ui0|+Δt1+θΔt⋅λih∑m-1i=1fik,
also we deduce that
(3.20)‖uim‖∞<‖ui0‖∞+‖Δt1+θΔt⋅λih‖∞∑m-1i=1‖fik‖∞.Remark 3.1 (cf. [4]).
We assume that the coerciveness assumption (Theorem2.3) is satisfied with λ=0, and for each k=1,…,n, we find
(3.21)‖uhk‖22+2Δt∑nk=1a(uhθ,k,uhθ,k)≤C(n)(∑nk=1Δt‖fθ,k‖22).


## 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
(3.1)(∂uh∂t,vh-uh)+a(uh,vh-uh)≥(f,vh-uh),vh∈Vh.Now, we apply theθ-scheme on the semidiscrete problem (3.1); for any θ∈[0,1] and k=1,…,n, we have
(3.2)(uhk-uhk-1,vh-uhθ,k)+Δt⋅a(uhθ,k,vh-uhθ,k)≥Δt⋅(fθ,k,vh-uhθ,k),vh∈Vh,
where
(3.3)uhθ,k=θuhk+(1-θ)uhk-1,fθ,k=θf(tk)+(1-θ)f(tk-1).It is possible to analyze the stability by taking the advantage of the structure of eigenvalues of the bilinear forma(·,·). 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
(3.4)ωj∈L2,ωj≠0:a(ωj,vh)=λj(ωj,vh),∀vh∈Vh.
The corresponding eigenfunctions {ωj} form a complete orthonormal basis in L2(Ω). In analogous way, when considering the finite dimensional problem in Wh, we find a sequence of eigenvalues δ≤λ1h≤λ2h≤⋯≤λm(h) and L2-orthonormal basis of eigenvectorss ωih∈Wh,i=1,2,…,m(h). Any function vh in Vh can thus be expanded with respect to the system ωih as
(3.5)vh=∑i=1m(h)(vh,ωih)ωih,
in particular, we have
(3.6)uhk=∑i=1m(h)uikωih,uik=(uhk,ωih).
Moreover, let fhk be the L2-orthogonal projection of fθ,k into Wh, that is, fhk∈Wh and
(3.7)(fhk,vh)=(fθ,k,vh),
and set
(3.8)fhk=∑i=1m(h)fikωih,fik=(fhk,ωih).
We are now in a position to prove the stability for θ∈[0,1/2[Choosing in (3.1) vh=0, thus we have
(3.9)1Δt(uhk-uhk-1,uhθ,k)+a(uhθ,k,uhθ,k)≤(fθ,k,uhθ,k),uhθ,k∈Vh.The inequalities (3.2) is equivalent to
(3.10)1Δt(uik-uik-1)+λih(θuik+(1-θ)uik-1)≤fik.Sinceωih are the eigenfunctions means
(3.11)a(ωih,ωih)=λih(ωih,ωih)=λih⋅δii=λih,
for each k=0,…,m(h)-1, we can rewrite (3.9) as
(3.12)uik≤1-(1-θ)⋅Δt⋅λih1+θΔt⋅λihuik-1+Δt1+θΔt⋅λihfik,
this inequality system stable if and only if
(3.13)|1-(1-θ)⋅Δt⋅λih1+θΔt⋅λih|<1,
that is to say
(3.14)2θ-1>-2λih⋅Δt
means
(3.15)Δt<2(1-2θ)λih.So that this relation satisfied for all the eigenvaluesλih of bilinear form a(·,·), we have to choose their highest value, and we take it for λmh=ρ(A) (rayon spectral)We deduce that ifθ≥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)Δt<2(1-2θ)ρ(A).We can prove that there exist two positive constantsc1,c2 such that
(3.17)c1h2≤λmh=c2h-2,
thus the method of θ-scheme is stable if and only if
(3.18)Δt<2C(1-2θ)h2.Notice that this condition is always satisfied if0≤θ<1/2. Hence, taking the absolute value of (3.12), we have
(3.19)|uim|<|ui0|+Δt1+θΔt⋅λih∑m-1i=1fik,
also we deduce that
(3.20)‖uim‖∞<‖ui0‖∞+‖Δt1+θΔt⋅λih‖∞∑m-1i=1‖fik‖∞.Remark 3.1 (cf. [4]).
We assume that the coerciveness assumption (Theorem2.3) is satisfied with λ=0, and for each k=1,…,n, we find
(3.21)‖uhk‖22+2Δt∑nk=1a(uhθ,k,uhθ,k)≤C(n)(∑nk=1Δt‖fθ,k‖22).


## 4. Asymptotic Behavior ofθ-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 constantC independent of h such that
(4.1)‖u∞-uh∞‖∞≤Ch2|log⁡h|2,
where u∞ and uh∞ are, respectively, stationery solutions to the following continue and discrete inequalities:
(4.2)b(u∞,v-u∞)≥(f+λu∞,v-u∞),v∈H01(Ω),(4.3)b(uh∞,vh-uh∞)≥(f+λu∞,vh-uh∞),vh∈Vh,
such that
(4.4)b(⋅,⋅)=a(⋅,⋅)+λ(⋅,⋅),where λ is a positive constant arbitrary.We have uhθ,k=θuhk+(1-θ)uhk-1≤θrhψ+(1-θ)rhψ=rhψ.Thus, we can rewrite (3.1) as, for uhθ,k∈Vh(4.5)(uhθ,kθΔt,vh-ũhk)+a(uhθ,k,vh-uhθ,k)≥(fθ,k+uhk-1θΔt,vh-uhθ,k),vh∈Vh.
Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities:
(4.6)b(uhθ,k,vh-ũhk)≥(fθ,k+μuhk-1,vh-uhθ,k),vh∈Vh,
such that
(4.7)b(uhθ,k,vh-uhθ,k)=μ(uhθ,k,vh-uhθ,k)+a(uhθ,k,vh-uhθ,k),vh,uhθ,k∈Vh,μ=1θΔt=Tθk,
where uhθ,1 is the solution to the following discrete inequality:
(4.8)a(uhθ,1,vh-uhθ,1)=(g(tk),vh-uhθ,1),vh∈Vh,
where g(tk) is a regular function given.


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

We consider the mapping(4.9)Th:L+∞(Ω)⟶Vhw⟶Th(w)=ξh,
where ξh is the unique solution of the following P.V.I: find ξh∈Vh(4.10)b(ξh,vh-ξh)≥(fθ,k+μw,vh-ξh),vh∈Vh.Proposition 4.2.
Under the previous hypotheses and notations, if one setsθ≥1/2, the mapping Th is a contraction in L∞(Ω) with rate of contraction 1/(1+β·θ·Δt). Therefore, Th admits a unique fixed point which coincides with the solution of P.V. I (4.7).Proof.
Forw, w̃ in L∞(Ω), we consider ξh=Th(w)=∂(fθ,k+μw,rhψ) and ξ̃h=Th(w̃)=∂(fθ,k+μw̃,rhψ̃) solution to quasivariational inequalities (4.7) with right-hand side Fθ,k=fθ,k+μwh,F̃θ,k=fθ,k+μw̃h.
Now, setting(4.11)ϕ=1β+μ‖Fθ,k-F̃θ,k‖∞,
then for ξh+ϕ is solution of
(4.12)b(ξh+ϕ,(vh+ϕ)-(ξh+ϕ))≥(Fθ,k+a0ϕ,(vh+ϕ)-(ξh+ϕ)),ξ+ϕ≤rhψ+ϕ,vh+ϕ≤rhψ+ϕ,∀vh∈Vh.
Also, we have
(4.13)Fθ,k≤F̃θ,k+‖Fθ,k-F̃θ,k‖∞≤F̃θ,k+a0β+μ‖Fθ,k-F̃θ,k‖∞≤F̃θ,k+a0ϕ,
thus
(4.14)∂h(Fθ,k,rhψ+ϕ)≤∂h(F̃θ,k+a0(x)ϕ,rhψ̃+ϕ)≤∂h(F̃θ,k,rψ̃)+ϕ,
hence
(4.15)ξh≤ξ̃h+ϕ.
Similarly, interchanging the roles of w and w̃, we also get
(4.16)ξ̃h≤ξh+ϕ.
Finally, this yields
(4.17)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤1β+μ‖Fθ,k-F̃θ,k‖∞≤1β+μ‖fθ,k+μw-fθ,k-μw̃‖∞‖w-w̃‖∞≤11+βθΔt‖w-w̃‖∞,
which completes the proof.Remark 4.3.
If we set0≤θ<1/2, the mapping Th 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 theθ-scheme is stable if and only if Δt<(2C/(1-2θ))h2.
Thus it can be easily show that(4.18)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤11+βθΔt‖w-w̃‖∞≤22+βθ(1-2θ)ρ(A)‖w-w̃‖∞≤11+βθ((1-2θ)/2Ch2)‖w-w̃‖∞=2Ch22Ch2+βθ(1-2θ)‖w-w̃‖∞,
also it can be found that
(4.19)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤11+βθ((1-2θ)/2Ch2)‖w-w̃‖∞=2Ch22Ch2+βθ(1-2θ)‖w-w̃‖∞,
thus the mapping Th is a contraction in L∞(Ω) with rate of contraction (2Ch2)/(2Ch2+βθ(1-2θ)). Therefore, Th admits a unique fixed point which coincides with the solution of P.V.I (4.7)
(4.20)‖Th(w)-Th(w̃)‖∞≤1β+μ‖Fθ,k-F̃θ,k‖∞=1β+μ‖fθ,k+μw-fθ,k-μw̃‖∞≤μβ+μ‖w-w̃‖∞≤11+βθΔt‖w-w̃‖∞.
This completes the proof.


### 4.2. Discrete Algorithm

Starting fromuh0=u0h (initial data) and the uhθ,1 solution of problem (4.7), we introduce the following discrete algorithm:
(4.21)uhθ,k=Thuhk-1,k=1,…,n,
where uhθ,k is the solution of the problem (4.7).Remark 4.4.
If we chooseθ=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: ifθ≥1/2(4.22)‖uhθ,k-uh∞‖∞≤(11+βθΔt)k‖uh∞-uh0‖∞,
and one has for
(4.23)‖uhθ,k-uh∞‖∞≤(2Ch22Ch2+βθ(1-2θ))k‖uh∞-uh0‖∞for0≤θ<12.Proof.
we set a first caseθ≥1/2, and we have
(4.24)uh∞=Thuh∞,‖uhθ,1-uh∞‖∞=‖Thuh0-Thuh∞‖∞≤(11+βθΔt)‖uh0-uh∞‖∞.
for k≥2, we use the Bensoussan-Lions' algorithm (uhk=Thuhk-1,k=1,…,n) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that(4.25)‖uhθ,k-uh∞‖∞≤(11+βθΔt)k‖uh0-uh∞‖∞,
so
(4.26)‖uhθ,k+1-uh∞‖∞=‖Thuhk-Thuh∞‖∞≤(11+βθΔt)‖uhk-uh∞‖∞,
thus
(4.27)‖uhθ,k+1-uh∞‖∞≤(11+βθΔt)k+1‖uh0-uh∞‖∞,
for a second case 0≤θ<1/2, it can be easily shown that
(4.28)‖uhθ,k-uh∞‖∞≤(2Ch22Ch2+βθ(1-2θ))k‖uh0-uh∞‖∞.


#### 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 inL∞-norm for parabolic variational inequalitiesNow, we evaluate the variation inL∞ between uhθ(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 Theorem4.1 and Proposition 4.5, one has  for the first case θ≥1/2,
(4.29)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(11+βθΔt)n],
and for the second case 0≤θ<1/2,
(4.30)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(2Ch22Ch2+βθ(1-2θ))n],
where C is a constant independent of h and k.Proof.
We have(4.31)uhθ,k(x)=uh(t,x)fort∈](k-1)Δt;kΔt[,
thus
(4.32)uhθ,n(x)=uh(T,x),
then
(4.33)‖uhθ(T,x)-u∞‖L∞(Ω)=‖uhθ,n-u∞‖L∞(Ω)≤‖uhθ,n-uh∞‖L∞(Ω)+‖uh∞-u∞‖L∞(Ω).
Using, Theorem 4.1 and Proposition 4.5, we have for θ≥1/2,
(4.34)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(11+βθΔt)n],
and for 0≤θ<1/2, we have
(4.35)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(2Ch22Ch2+βθ(1-2θ))n].


## 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 constantC independent of h such that
(4.1)‖u∞-uh∞‖∞≤Ch2|log⁡h|2,
where u∞ and uh∞ are, respectively, stationery solutions to the following continue and discrete inequalities:
(4.2)b(u∞,v-u∞)≥(f+λu∞,v-u∞),v∈H01(Ω),(4.3)b(uh∞,vh-uh∞)≥(f+λu∞,vh-uh∞),vh∈Vh,
such that
(4.4)b(⋅,⋅)=a(⋅,⋅)+λ(⋅,⋅),where λ is a positive constant arbitrary.We have uhθ,k=θuhk+(1-θ)uhk-1≤θrhψ+(1-θ)rhψ=rhψ.Thus, we can rewrite (3.1) as, for uhθ,k∈Vh(4.5)(uhθ,kθΔt,vh-ũhk)+a(uhθ,k,vh-uhθ,k)≥(fθ,k+uhk-1θΔt,vh-uhθ,k),vh∈Vh.
Thus, our problem (4.5) is equivalent to the following noncoercive elliptic variational inequalities:
(4.6)b(uhθ,k,vh-ũhk)≥(fθ,k+μuhk-1,vh-uhθ,k),vh∈Vh,
such that
(4.7)b(uhθ,k,vh-uhθ,k)=μ(uhθ,k,vh-uhθ,k)+a(uhθ,k,vh-uhθ,k),vh,uhθ,k∈Vh,μ=1θΔt=Tθk,
where uhθ,1 is the solution to the following discrete inequality:
(4.8)a(uhθ,1,vh-uhθ,1)=(g(tk),vh-uhθ,1),vh∈Vh,
where g(tk) is a regular function given.


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

We consider the mapping(4.9)Th:L+∞(Ω)⟶Vhw⟶Th(w)=ξh,
where ξh is the unique solution of the following P.V.I: find ξh∈Vh(4.10)b(ξh,vh-ξh)≥(fθ,k+μw,vh-ξh),vh∈Vh.Proposition 4.2.
Under the previous hypotheses and notations, if one setsθ≥1/2, the mapping Th is a contraction in L∞(Ω) with rate of contraction 1/(1+β·θ·Δt). Therefore, Th admits a unique fixed point which coincides with the solution of P.V. I (4.7).Proof.
Forw, w̃ in L∞(Ω), we consider ξh=Th(w)=∂(fθ,k+μw,rhψ) and ξ̃h=Th(w̃)=∂(fθ,k+μw̃,rhψ̃) solution to quasivariational inequalities (4.7) with right-hand side Fθ,k=fθ,k+μwh,F̃θ,k=fθ,k+μw̃h.
Now, setting(4.11)ϕ=1β+μ‖Fθ,k-F̃θ,k‖∞,
then for ξh+ϕ is solution of
(4.12)b(ξh+ϕ,(vh+ϕ)-(ξh+ϕ))≥(Fθ,k+a0ϕ,(vh+ϕ)-(ξh+ϕ)),ξ+ϕ≤rhψ+ϕ,vh+ϕ≤rhψ+ϕ,∀vh∈Vh.
Also, we have
(4.13)Fθ,k≤F̃θ,k+‖Fθ,k-F̃θ,k‖∞≤F̃θ,k+a0β+μ‖Fθ,k-F̃θ,k‖∞≤F̃θ,k+a0ϕ,
thus
(4.14)∂h(Fθ,k,rhψ+ϕ)≤∂h(F̃θ,k+a0(x)ϕ,rhψ̃+ϕ)≤∂h(F̃θ,k,rψ̃)+ϕ,
hence
(4.15)ξh≤ξ̃h+ϕ.
Similarly, interchanging the roles of w and w̃, we also get
(4.16)ξ̃h≤ξh+ϕ.
Finally, this yields
(4.17)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤1β+μ‖Fθ,k-F̃θ,k‖∞≤1β+μ‖fθ,k+μw-fθ,k-μw̃‖∞‖w-w̃‖∞≤11+βθΔt‖w-w̃‖∞,
which completes the proof.Remark 4.3.
If we set0≤θ<1/2, the mapping Th 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 theθ-scheme is stable if and only if Δt<(2C/(1-2θ))h2.
Thus it can be easily show that(4.18)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤11+βθΔt‖w-w̃‖∞≤22+βθ(1-2θ)ρ(A)‖w-w̃‖∞≤11+βθ((1-2θ)/2Ch2)‖w-w̃‖∞=2Ch22Ch2+βθ(1-2θ)‖w-w̃‖∞,
also it can be found that
(4.19)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤11+βθ((1-2θ)/2Ch2)‖w-w̃‖∞=2Ch22Ch2+βθ(1-2θ)‖w-w̃‖∞,
thus the mapping Th is a contraction in L∞(Ω) with rate of contraction (2Ch2)/(2Ch2+βθ(1-2θ)). Therefore, Th admits a unique fixed point which coincides with the solution of P.V.I (4.7)
(4.20)‖Th(w)-Th(w̃)‖∞≤1β+μ‖Fθ,k-F̃θ,k‖∞=1β+μ‖fθ,k+μw-fθ,k-μw̃‖∞≤μβ+μ‖w-w̃‖∞≤11+βθΔt‖w-w̃‖∞.
This completes the proof.


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

We consider the mapping(4.9)Th:L+∞(Ω)⟶Vhw⟶Th(w)=ξh,
where ξh is the unique solution of the following P.V.I: find ξh∈Vh(4.10)b(ξh,vh-ξh)≥(fθ,k+μw,vh-ξh),vh∈Vh.Proposition 4.2.
Under the previous hypotheses and notations, if one setsθ≥1/2, the mapping Th is a contraction in L∞(Ω) with rate of contraction 1/(1+β·θ·Δt). Therefore, Th admits a unique fixed point which coincides with the solution of P.V. I (4.7).Proof.
Forw, w̃ in L∞(Ω), we consider ξh=Th(w)=∂(fθ,k+μw,rhψ) and ξ̃h=Th(w̃)=∂(fθ,k+μw̃,rhψ̃) solution to quasivariational inequalities (4.7) with right-hand side Fθ,k=fθ,k+μwh,F̃θ,k=fθ,k+μw̃h.
Now, setting(4.11)ϕ=1β+μ‖Fθ,k-F̃θ,k‖∞,
then for ξh+ϕ is solution of
(4.12)b(ξh+ϕ,(vh+ϕ)-(ξh+ϕ))≥(Fθ,k+a0ϕ,(vh+ϕ)-(ξh+ϕ)),ξ+ϕ≤rhψ+ϕ,vh+ϕ≤rhψ+ϕ,∀vh∈Vh.
Also, we have
(4.13)Fθ,k≤F̃θ,k+‖Fθ,k-F̃θ,k‖∞≤F̃θ,k+a0β+μ‖Fθ,k-F̃θ,k‖∞≤F̃θ,k+a0ϕ,
thus
(4.14)∂h(Fθ,k,rhψ+ϕ)≤∂h(F̃θ,k+a0(x)ϕ,rhψ̃+ϕ)≤∂h(F̃θ,k,rψ̃)+ϕ,
hence
(4.15)ξh≤ξ̃h+ϕ.
Similarly, interchanging the roles of w and w̃, we also get
(4.16)ξ̃h≤ξh+ϕ.
Finally, this yields
(4.17)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤1β+μ‖Fθ,k-F̃θ,k‖∞≤1β+μ‖fθ,k+μw-fθ,k-μw̃‖∞‖w-w̃‖∞≤11+βθΔt‖w-w̃‖∞,
which completes the proof.Remark 4.3.
If we set0≤θ<1/2, the mapping Th 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 theθ-scheme is stable if and only if Δt<(2C/(1-2θ))h2.
Thus it can be easily show that(4.18)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤11+βθΔt‖w-w̃‖∞≤22+βθ(1-2θ)ρ(A)‖w-w̃‖∞≤11+βθ((1-2θ)/2Ch2)‖w-w̃‖∞=2Ch22Ch2+βθ(1-2θ)‖w-w̃‖∞,
also it can be found that
(4.19)‖∂h(Fθ,k,rhψ)-∂h(F̃θ,k,rhψ̃)‖∞≤11+βθ((1-2θ)/2Ch2)‖w-w̃‖∞=2Ch22Ch2+βθ(1-2θ)‖w-w̃‖∞,
thus the mapping Th is a contraction in L∞(Ω) with rate of contraction (2Ch2)/(2Ch2+βθ(1-2θ)). Therefore, Th admits a unique fixed point which coincides with the solution of P.V.I (4.7)
(4.20)‖Th(w)-Th(w̃)‖∞≤1β+μ‖Fθ,k-F̃θ,k‖∞=1β+μ‖fθ,k+μw-fθ,k-μw̃‖∞≤μβ+μ‖w-w̃‖∞≤11+βθΔt‖w-w̃‖∞.
This completes the proof.


## 4.2. Discrete Algorithm

Starting fromuh0=u0h (initial data) and the uhθ,1 solution of problem (4.7), we introduce the following discrete algorithm:
(4.21)uhθ,k=Thuhk-1,k=1,…,n,
where uhθ,k is the solution of the problem (4.7).Remark 4.4.
If we chooseθ=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: ifθ≥1/2(4.22)‖uhθ,k-uh∞‖∞≤(11+βθΔt)k‖uh∞-uh0‖∞,
and one has for
(4.23)‖uhθ,k-uh∞‖∞≤(2Ch22Ch2+βθ(1-2θ))k‖uh∞-uh0‖∞for0≤θ<12.Proof.
we set a first caseθ≥1/2, and we have
(4.24)uh∞=Thuh∞,‖uhθ,1-uh∞‖∞=‖Thuh0-Thuh∞‖∞≤(11+βθΔt)‖uh0-uh∞‖∞.
for k≥2, we use the Bensoussan-Lions' algorithm (uhk=Thuhk-1,k=1,…,n) for a noncoercive elliptic quasivariational inequalities (cf., e.g., [2, 3]) for details.
We assume that(4.25)‖uhθ,k-uh∞‖∞≤(11+βθΔt)k‖uh0-uh∞‖∞,
so
(4.26)‖uhθ,k+1-uh∞‖∞=‖Thuhk-Thuh∞‖∞≤(11+βθΔt)‖uhk-uh∞‖∞,
thus
(4.27)‖uhθ,k+1-uh∞‖∞≤(11+βθΔt)k+1‖uh0-uh∞‖∞,
for a second case 0≤θ<1/2, it can be easily shown that
(4.28)‖uhθ,k-uh∞‖∞≤(2Ch22Ch2+βθ(1-2θ))k‖uh0-uh∞‖∞.


### 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 inL∞-norm for parabolic variational inequalitiesNow, we evaluate the variation inL∞ between uhθ(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 Theorem4.1 and Proposition 4.5, one has  for the first case θ≥1/2,
(4.29)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(11+βθΔt)n],
and for the second case 0≤θ<1/2,
(4.30)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(2Ch22Ch2+βθ(1-2θ))n],
where C is a constant independent of h and k.Proof.
We have(4.31)uhθ,k(x)=uh(t,x)fort∈](k-1)Δt;kΔt[,
thus
(4.32)uhθ,n(x)=uh(T,x),
then
(4.33)‖uhθ(T,x)-u∞‖L∞(Ω)=‖uhθ,n-u∞‖L∞(Ω)≤‖uhθ,n-uh∞‖L∞(Ω)+‖uh∞-u∞‖L∞(Ω).
Using, Theorem 4.1 and Proposition 4.5, we have for θ≥1/2,
(4.34)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(11+βθΔt)n],
and for 0≤θ<1/2, we have
(4.35)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(2Ch22Ch2+βθ(1-2θ))n].


## 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 inL∞-norm for parabolic variational inequalitiesNow, we evaluate the variation inL∞ between uhθ(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 Theorem4.1 and Proposition 4.5, one has  for the first case θ≥1/2,
(4.29)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(11+βθΔt)n],
and for the second case 0≤θ<1/2,
(4.30)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(2Ch22Ch2+βθ(1-2θ))n],
where C is a constant independent of h and k.Proof.
We have(4.31)uhθ,k(x)=uh(t,x)fort∈](k-1)Δt;kΔt[,
thus
(4.32)uhθ,n(x)=uh(T,x),
then
(4.33)‖uhθ(T,x)-u∞‖L∞(Ω)=‖uhθ,n-u∞‖L∞(Ω)≤‖uhθ,n-uh∞‖L∞(Ω)+‖uh∞-u∞‖L∞(Ω).
Using, Theorem 4.1 and Proposition 4.5, we have for θ≥1/2,
(4.34)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(11+βθΔt)n],
and for 0≤θ<1/2, we have
(4.35)‖uhθ,n-u∞‖∞≤C[h2|log⁡h|2+(2Ch22Ch2+βθ(1-2θ))n].


## 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.



---

*Source: 703670-2011-07-07.xml*