On finite element approximation of system of

Bencheikh Le Hocine et al., Cogent Mathematics (2016), 3: 1251386 http://dx.doi.org/10.1080/23311835.2016.1251386

COMPUTATIONAL SCIENCE | RESEARCH ARTICLE

On finite element approximation of system of parabolic quasi-variational inequalities related to stochastic control problems Received: 10 August 2016 Accepted: 14 October 2016 First Published: 25 October 2016 *Corresponding author: Salah Boulaaras, Department of Mathematics, College of Sciences and Arts, Al-Ras, Qassim University, Buraydah, Kingdom of Saudi Arabia; Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria E-mail: [email protected] Reviewing editor: Dumitru Baleanu, Cankaya University, Turkey Additional information is available at the end of the article

Mohamed Amine Bencheikh Le Hocine1,2, Salah Boulaaras3,4* and Mohamed Haiour2

Abstract: In this paper, an optimal error estimate for system of parabolic quasivariational inequalities related to stochastic control problems is studied. Existence and uniqueness of the solution is provided by the introduction of a constructive algorithm. An optimally L∞-asymptotic behavior in maximum norm is proved using the semiimplicit time scheme combined with the finite element spatial approximation. The approach is based on the concept of subsolution and discrete regularity. Subjects: Advanced Mathematics; Applied Mathematics; Mathematics & Statistics; Science Keywords: parabolic quasi-variational inequalities; Hamilton–Jacobi–Bellman equation; finite element methods; subsolutions method; L∞-asymptotic behavior; orthogonal polynomials and special functions 1991 Mathematics subject classification: 35 R 35; 49 J 40 1. Introduction We consider the following Parabolic Quasi-Variational Inequalities (PQVI):

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Mohamed Amine Bencheikh Le Hocine

𝜕ui + 𝔸ui − f i ≤ 0, ui ≤ Mui , �� 𝜕t i � 𝜕u + 𝔸ui − f i ui − Mui = 0, in ℚT : = Ω ×]0, T[ ; 𝜕t � ui �t=0 = ui0 , ∀i = 1, 2, … , J, in Ω; � ∑ ui = 0, on T : = ]0, T[×Γ ,

(1.1)

ABOUT THE AUTHORS

PUBLIC INTEREST STATEMENT

Mohamed Amine Bencheick Le Hocine is an associate professor at Tamanarast University. He received his PhD on Numerical Analysis in January 2014 from University of Annaba, Algeria. He published more than 25 papers in international refereed journals. Salah Boulaaras was born in 1985 in Algeria. He received his PhD in January 2012 He serves as an associate Professor at Qassim University, KSA. He published more than 25 papers in international refereed journals. Mohamed Haiour is a full Professor at Annaba University. He received his PhD in Numerical Analysis in 2004 from the University of Annaba. He has more than 34 publications in refereed journal and conference papers.

The stationary and evolutionary free boundary problems are accomplished in some applications; for example, in stochastic control, their solution characterize the in mum of the cost function associated to an optimally controlled stochastic switching process without costs for switching and for the calculus of quasi-stationary state for the simulation of petroleum or gaseous deposit.

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

Page 1 of 20


where Ω is a bounded convex domain in ℝ , d ≥ 1 with smooth boundary Γ, and ℚT set in ℝ × ℝ,

[

d

]

ℚT = :Ω × 0, T with T < +∞,

∑

T

the form

�

�

d

i

= :Γ × 0, T , 𝔸 are second order, uniformly elliptic operators of

∑ i 𝜕 𝜕2 + bj (x) + ai0 (x), (1.2) 𝜕x 𝜕x 𝜕x j k j j=1 j,k=1 ( ) i i i 2 ̄ ̄ 1 ≤ j, k ≤ d are sufficiently smooth coefficients , x ∈ Ω, where ∀i = 1, ..., J, ajk , bj , a0 ∈ C Ω

𝔸i = −

d ∑

d

aijk (x)

and satisfy the following conditions:

aijk (x) = aikj (x); ai0 (x) ≥ 𝛽 > 0,

(1.3)

𝛽 is a constant

and

∑n

ai (x)𝜉j 𝜉k j,k=1 jk

≥ 𝛾|𝜉|2 ;

𝜉 ∈ ℝd ,

𝛾 > 0,

̄ x ∈ Ω.

(1.4)

f i are given functions satisfying the following condition ( ) f i ∈ L2 0, T, L∞ (Ω) ∩ C 1 (0, T, H−1 (Ω)), and f i ≥ 0.

(1.5)

Mui represents the obstacle of stochastic control defined by: (1.6)

Mui = k + ui+1 where k is a strictly positive constant.

This problem arises in stochastic control problems. It also plays a fundamental role in solving the Hamilton–Jacobi–Bellman equation (Evans & Friedman, 1979; Lions & Menaldi, 1979). In this paper, we are concerned with the numerical approximation in the L norm for the problem (1.1). From Lions and Menaldi (1979), we know that (1.1) can be approximated by ( the following sys) ∞

1 2 M )J quasi-variational inequalities (PQVIs): find a vector U = u , u , ..., u ∈ (tem( of parabolic ) L2 0, T, H01 (Ω) such that

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

� � � � � � 𝜕 u i , v − u i + ai u i , v − u i ≥ f i , v − u i , 𝜕t ui ≤ k + ui+1 , v ≤ k + ui+1 , uJ+1 = u1 ; � � ui 0 = ui0 ,

∀v ∈ H01 (Ω); i = 1, 2, ..., J;

(1.7)

where a(u, v) is a continuous and noncoercive bilinear form associated with elliptic operator 𝔸 1 defined as: for any u, v ∈ H (Ω)

a(u, v) =

∫Ω

) d ∑ 𝜕u 𝜕u 𝜕v + b (x) v + a0 (x)uv dx ajk (x) 𝜕 xj 𝜕 xk k=1 k 𝜕xk j,k=1

(

d ∑

(1.8)

2

and (.,.) is the inner product in L (Ω). Next we give consideration to a discrete version of (1.1): let 𝜏h be a regular and quasi-uniform triangulation of Ω; h > 0 is the mesh size. Let also 𝕍h be the finite element space consisting of continu{ } ous piecewise linear functions vanishing on Γ, 𝜑i , i = 1, ..., m(h) the basis functions of 𝕍h, and rh the (usual restriction ) operator. We consider the fully discretized problem: find ( )M Uhn = u1,n , u2,n , ..., uM,n ∈ 𝕍h such that for all n = 1, 2, ..., N h h h Page 2 of 20


� � ⎧ ui,n − ui,n−1 � � � � h h i,n i,n i,n i,n ⎪ , v − u , v − u , vh − uh + ai ui,n ; ≥ f h h h h h ⎪ △t � � � � ⎪ i,n i+1,n , vh ≤ rh k + ui+1,n ; ⎨ uh ≤ rh k + uh h ⎪ M+1,n 1,n ⎪ uh � �= uh ; ⎪ ui,n 0 = ui,n , h,0 ⎩ h with △t: =

∀vh ∈ 𝕍h ; (1.9)

( ) T i an appropriate approximation of u0 . ; tn = n △ t the time step, f i,n = f i tn and ui,n h,0 N

Error estimates for piecewise linear finite element approximations of parabolic variational and quasi-variational inequalities have been established in various papers (cf. e.g. Achdou, Hecht, & Pommier, 2008; Alfredo, 1987; Bencheikh Le Hocine, Boulaaras, & Haiour, 2016; Bensoussan & Lions, 1973; Berger & Falk, 1977; Boulaaras, Bencheikh Le Hocine, & Haiour, 2014; Diaz & Defonso, 1985; Scarpini & Vivaldi, 1977). More recently, Bencheikh Le Hocine and Haiour (2013) exploited the above arguments for system of parabolic quasi-variational inequalities, where they analyzed the semiimplicit Euler scheme with respect to the t- variable combined with a finite element spatial approxi∞ mation and gave (for d ≥ 1) the following L -asymptotic behavior:

( ( )n ) 1 ‖U (T, .) − U∞ (.)‖ ≤ C h2 |log h|4 + ‖ h ‖∞ | | 1 + 𝛽Δt

The quasi-optimal L -asymptotic behavior (d ≥ 1): for 𝜃 ≥ ∞

(1.10) 1 2

( ( )n ) 1 |U (T, .) − U∞ (.)‖ ≤ C h2 |log h|3 + | h ‖∞ | | 1 + 𝛽𝜃Δt and for 𝜃 ∈ [ 0,

(1.11)

1 [ 2

(

‖U (T, .) − U (.)‖ ≤ C h log h + ‖ h ‖∞ | | 2|

∞

|3

(

2Ch2 ( ) 2 2Ch + 𝛽𝜃 1 − 2𝜃 𝜌(A)

)n ) ,

(1.12)

‖ ( i )‖

‖ where 𝜌(𝔸) = min‖ ‖𝜌 𝔸 ‖ is the spectral radius of the elliptic operator 𝔸, has been obtained in 1≤i≤J ‖

‖∞

Boulaaras and Haiour (2014).

In the current paper, we shall employ the concepts of subsolutions and discrete regularity (Bencheikh Le Hocine et al., 2016; Boulbrachene, 2014, 2015a, 2015b; Cortey-Dumont, 1987, 1985). More precisely, we use the characterization the continuous solution (resp. the discrete solution) as the maximum elements of the set of continuous subsolutions (resp. the maximum elements of the ∞ set of discrete subsolutions), in order to yield the following optimal L -asymptotic behavior (for d ≥ 1):

( ( )n ) 1 ‖U (T, .) − U∞ (.)‖ ≤ C h2 |log h|2 + . ‖ h ‖∞ | | 1 + 𝛽Δt

(1.13)

The paper is organized as follows. In Section 2, we present the continuous problem and study some qualitative properties. The discrete problem is proposed in Section 3. In Section 4, we derive ∞ an L –error estimate of the approximation. The main result of the paper is presented in Section 5.

Page 3 of 20


2. Statement of the continuous system 2.1. Existence and uniqueness 2.1.1. The time discretization We discretize the problem (1.1) or (1.7) with respect to time by using the semi-implicit scheme. ( ) i,n 1 i Therefore, we search a sequence of elements u ∈ H0 (Ω), 1 ≤ i ≤ J, which approaches u tn , tn = k Δt, with initial data ui,0 = ui,0. Thus, we have for n = 1, ..., N

� � � � � � ⎧ ui,n − ui,n−1 i,n i i,n i,n i,n i,n + a u , v − u f , v − u , , v − u ≥ ⎪ ⎪ i,n △t i+1,n , v ≤ k + ui+1,n ; ⎨ u ≤k+u ⎪ uJ+1,n = u1,n ; ⎪ i� � i ⎩ u 0 = u0 ,

∀v ∈ H01 (Ω); (2.1)

where

△t =

T . N (

(2.2)

)

u , v − ui,n to both parties of the inequalities (2.1), we get △t � � � � � � ui,n−1 1 ui,n , v − ui,n ≥ f i,n + , v − ui,n ; ai ui,n , v − ui,n + △t △t ui,n ≤ k + ui+1,n , v ≤ k + ui+1,n ; uJ+1,n = u1,n ; � � ui 0 = ui0 .

By adding

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

i,n−1

(2.3)

1

The bilinear form a(., .), being noncoercive in H0 (Ω), there exist two constants 𝛼 > 0 and 𝜆 > 0 such that:

a(𝜑, 𝜑) + 𝜆‖𝜑‖2L2 (Ω) ≥ 𝛼‖𝜑‖2H1 (Ω) for all 𝜑 ∈ H01 (Ω). 0

(2.4)

Set (2.5)

b(u, v) = a(u, v) + 𝜆(u, v).

Then the bilinear form b(., .) is strongly coercive and therefore, the continuous the problem (2.3) ( 1,n ) ( 1 )J n J,n reads as follows: find U = u , ..., u ∈ H0 (Ω) such that for all n = 1, ..., N

� � � � ⎧ bi ui,n , v − ui,n ≥ f i,n + 𝜆ui,n−1 , v − ui,n , ⎪ ⎨ ui,n ≤ k + ui+1,n , v ≤ k + ui+1,n ; ⎪ uJ+1,n = u1,n , ⎩

∀v ∈ H01 (Ω); (2.6)

where

� � � � � ⎧ i � i,n i,n = ai ui,n , v − ui,n + 𝜆 ui,n , v − ui,n , ⎪ b u , v−u ⎨ 1 > 0. ⎪ 𝜆= Δt ⎩

(2.7)

Page 4 of 20


Remark 1 The problem (2.6) is called the coercive continuous problem of elliptic quasi-variational inequalities (QVI). Notation 1 We denote by u

(

0

1

J

i,n

( ) = 𝜎 f i,n ; k + ui+1,n the solution of problem (2.6).

)

Let U = U0 = u0 , ..., u0 be the solution of the following continuous equation:

( ) ( ) bi ui0 , v = f i + 𝜆ui0 , v ,

∀v ∈ H01 (Ω).

(2.8)

The existence and uniqueness of a continuous solution is obtained by means of Banach’s fixed point theorem.

2.1.2. A fixed point mapping associated with continuous system (2.6) Let ℍ = +

J ∏ ∞ ∞ L∞ + (Ω), where L+ (Ω) is the positive cone of L (Ω). We introduce the following mapping: i=1

𝕋 : ℍ+ ⟶ ℍ+ ,

( ) W → 𝕋 W = 𝜉 = 𝜉 1 , ..., 𝜉 J , ( ) i i i i+1 where 𝜉 = 𝜎 f + 𝜆.w ; k + 𝜉 ∈ H01 (Ω) solves the following coercive system of QVI: � � � � ⎧ bi 𝜉 i , v − 𝜉 i ≥ f i + 𝜆.w i , v − 𝜉 i ; ⎪ ⎨ 𝜉 i ≤ k + 𝜉 i+1 , v ≤ k + 𝜉 i+1 ; ⎪ 𝜉 J+1 = 𝜉 1 . ⎩

(2.9)

(2.10)

Theorem 1 Under the preceding hypotheses and notations, the mapping 𝕋 is a contraction in ℍ+ with 1 . Therefore, 𝕋 admits a unique fixed point which coincides with the a contraction constant 𝜌 = 𝛽Δt + 1 solution of problem (2.6). Proof Boulbrachene, Haiour, and Chentouf (2002), taking 𝜆 =

1 , we have: Δt

1 ‖ ‖ ̃‖ ̃‖ ‖𝕋 W − 𝕋 W ‖ ≤ ‖W − W ‖ , ‖ ‖∞ 𝛽Δt + 1 ‖ ‖∞

which completes the proof.

✷

The mapping 𝕋 clearly generates the following iterative scheme.

2.2. A continuous iterative scheme 0

Starting from U = U0, the solution of Equation (2.8), we define the sequence: (2.11)

Un = 𝕋 Un−1 , n

where U is a solution of the problem (2.6).

2.2.1. Geometrical convergence In what follows, we shall establish the geometrical convergence of the proposed iterative scheme. Proposition 1 Under conditions of Theorem 1, we have: ‖ ‖ max‖ui,n − ui,∞ ‖ ≤ ‖∞ 1≤i≤J ‖

(

1 𝛽Δt + 1

)n

‖ ‖ max‖ui0 − ui,∞ ‖ , ‖∞ 1≤i≤J ‖

(2.12)

Page 5 of 20


where U∞ is the asymptotic solution of the problem of quasi-variational inequalities: find ( ) ( )J U∞ = u1,∞ , ..., uJ,∞ ∈ H01 (Ω) such that � � � � ⎧ bi ui,∞ , v − ui,∞ ≥ f i + 𝜆.ui,∞ , v − ui,∞ , ⎪ i,∞ ⎨ u ≤ k + ui+1,∞ , v ≤ k + ui+1,∞ ; ⎪ J+1,∞ = u1,∞ . ⎩ u

v ∈ H01 (Ω);

(2.13)

Proof Under Theorem 1, we have for n = 1 ‖ i,1 ‖ ‖ ‖ ‖u − ui,∞ ‖ = ‖𝕋 ui0 − 𝕋 ui,∞ ‖ ‖∞ ‖ ‖∞ ‖ ) ( 1 ‖ ‖ i ≤ ‖u − ui,∞ ‖ ‖∞ 𝛽Δt + 1 ‖ 0

Now, we assume that ‖ i,n ‖ ‖u − ui,∞ ‖ ≤ ‖ ‖∞

(

1 𝛽Δt + 1

)n

‖ i ‖ ‖u0 − ui,∞ ‖ , ‖ ‖∞

then ‖ i,n+1 ‖ ‖ ‖ − ui,∞ ‖ = ‖𝕋 ui,n − 𝕋 ui,∞ ‖ ‖u ‖∞ ‖ ‖∞ ‖ ) ( 1 ‖ ‖ i,n ≤ ‖u − ui,∞ ‖ . ‖∞ 𝛽Δt + 1 ‖

Thus, ‖ i,n+1 ‖ − ui,∞ ‖ ≤ ‖u ‖ ‖∞ ≤

( (

1 𝛽Δt + 1 1 𝛽Δt + 1

) ( ⋅ )n+1

1 𝛽Δt + 1

)n

‖ i ‖ ‖u0 − ui,∞ ‖ ‖ ‖∞

‖ i ‖ ‖u0 − ui,∞ ‖ , ‖ ‖∞


✷

In what follows, we shall give monotonicity and Lipschitz dependence with respect to the righthand sides and parameter k for the solution of system (2.6). These properties together with the notion of subsolution will play a fundamental role in the study the error estimate between the nth iterates of the continuous system (2.6) and its discrete counterpart.

2.3. A monotonicity property (

Let k and k̃ be two parameters, f

(

1,n

J,n

)

(

1,n

) ( ) , ..., f J,n and f̃ 1,n , ..., f̃ J,n be two families of right-hand sides.

1,n

J,n

)

̃ to system of quasi-variational We denote u , ..., u (resp. u ũ ) ) the (corresponding solution ( 1,n , ..., ) J,n 1,n J,n inequalities (2.6) defined with f , ..., f ; k . (resp. f̃ , ..., f̃ ; k̃ . Then, we have the following Lemma 1 (cf. Boulbrachene et al., 2002) If f i,n ≥ f̃ i,n and k ≥ k̃ , then ui,n ≥ ũ i,n .

(2.14)

2.4. Lipschitz dependence with respect to the right-hand sides and the parameter k Proposition 2 Under conditions of Lemma 1, we have

Page 6 of 20


) ( ‖ | ‖ ‖ ‖ | max‖ui,n − ũ i,n ‖ ≤ C max |k − k̃ | + ‖f i,n − f̃ i,n ‖ , ‖∞ | ‖ ‖∞ 1≤i≤J | 1≤i≤J ‖

(2.15)

where C is a constant such that ai0 C ≥ 1.

(2.16)

Proof Let ( ) | ‖ ‖ | 𝜙i = C |k − k̃ | + ‖f i,n − f̃ i,n ‖ . | ‖ ‖∞ |

Then, from (2.16) it is easy to see that | | ‖ ‖ f̃ i,n ≤ f i,n + |k − k̃ | + ‖f i,n − f̃ i,n ‖ | | ‖ ‖∞ ( ) | | ‖ ‖ ≤ f i,n + ai0 C |k − k̃ | + ‖f i,n − f̃ i,n ‖ | | ‖ ‖∞ ≤ f i,n + ai0 𝜙i ,

and ) ( ‖ | ‖ | k̃ ≤ k + C |k − k̃ | + ‖f i,n − f̃ i,n ‖ ‖∞ | ‖ | ≤ k + 𝜙i .

So, due to Lemma 1 it follows that ( ) ( ) 𝜎 f̃ i,n ; k̃ + ũ i+1,n ≤ 𝜎 f i,n + ai0 𝜙i ; k + 𝜙i + ui+1,n ( ) ≤ 𝜎 f i,n ; k + ui+1,n + 𝜙i ,

hence ( ) ( ) 𝜎 f̃ i,n ; k̃ + ũ i+1,n − 𝜎 f i,n ; k + ui+1,n ≤ 𝜙i .

Interchanging the role of f i,n and f̃ i,n, k and k̃ we also get ( ) ( ) 𝜎 f i,n ; k + ui+1,n − 𝜎 f̃ i,n ; k̃ + ũ i+1,n ≤ 𝜙i .

Then ) ( )‖ ( ) ‖ ( i,n ‖ ‖𝜎 f ; k + ui+1,n − 𝜎 f̃ i,n ; k̃ + ũ i+1,n ‖ ≤ C ||k − k̃ || + ‖ ‖f i,n − f̃ i,n ‖ , ‖ ‖ | ‖ ‖∞ | ‖∞ ‖


✷

2.5. Characterization of the solution of system (2.6) as the envelope of continuous subsolutions ( ) ( )J Definition 1 Z = z1 , ..., zJ ∈ H01 (Ω) is said to be a continuous subsolution for the system of quasi-variational inequalities (2.6) if � � � � ⎧ bi zi , v ≤ f i + 𝜆.zi , v , ⎪ i ⎨ z ≤ k + zi+1 , v ≥ 0; ⎪ J+1 ⎩ z = z1 .

v ∈ H01 (Ω);

(2.17)

Page 7 of 20


Notation 2 Let 𝕏 denote the set of such subsolutions. Theorem 2 (cf. Bensoussan & Lions, 1978) The solution of (2.6) is the least upper bound of the set 𝕏.

3. Statement of the discrete system In this section we shall see that the discrete system below inherits all the qualitative properties of the continuous system, provided the discrete maximum principle assumption is satisfied. Their respective proofs shall be omitted, as they are very similar to their continuous analogues.

3.1. Spatial discretization ( )

Let Ms , 1 ≤ s ≤ m h denote the vertex of the triangulation 𝜏h, and let 𝜙l, 1 ≤ l ≤ m h , denote the functions of 𝕍h which satisfies:

( ) 𝜙l Ms = 𝛿ls ,

( )

1 ≤ l, s ≤ m(h).

( )) ( ) ( 1 1 ̄ So that the function 𝜙l from a basis of 𝕍h . ∀vh ∈ L 0, T; H0 (Ω) ∩ C 0, T; H0 Ω

(3.1)

2

m(h)

rh v =

∑ ( ) v Ml 𝜙l (x),

(3.2)

l=1

represents the interpolate of v over 𝜏h .

3.1.1. The discrete maximum principle (dmp) Denote by 𝔹 is the matrix with generic entries: ∀ i = 1, ..., J

( ) ( ) 𝔹il, s = bi 𝜙l , 𝜙s = ai 𝜙l , 𝜙s + 𝜆 𝜙l 𝜙s dx, �

( ) 1 ≤ l, s ≤ m h .

(3.3)

Ω

Lemma 2 (cf. Cortey-Dumont, 1983) The matrix 𝔹 is an M-matrix.

3.2. Existence and uniqueness

(

1

J

)

( )M

The discrete problem of PQVI consists of seeking Uh = uh , ..., uh ∈ 𝕍h

� � � � � � ⎧ 𝜕 u i , v − u i + ai u i , v − u i ≥ f i , v − u i , h h h h ⎪ 𝜕t h �h � �h � ⎪ ui ≤ r k + ui+1 , v ≤ r k + ui+1 ; h h h h h ⎨ h 1 ⎪ uJ+1 = u ; h ⎪ hi � � i ⎩ uh 0 = u0h ,

such that

∀vh ∈ 𝕍h ; (3.4)

or equivalently,

� � � � i,n i,n i,n i,n ⎧ bi ui,n , v − u f + 𝜆.u , v − u ; ≥ h h h � � h �h ⎪ i,n h � i+1,n i+1,n ⎨ uh ≤ rh k + uh , vh ≤ rh k + uh ; ⎪ J+1,n 1,n ⎩ uh = uh . (

Notation 3 We denote by uh = 𝜎h f i,n

0

(

1

M

i,n

(3.5)

)) ( the solution of system (3.5). ; rh k + ui+1,n h

)

Let Uh = U0h = u0h , ..., u0h be the solution of the following discrete equation:

( ) ( ) bi ui0h , vh = f i + 𝜆ui0h , vh ,

∀vh ∈ 𝕍h .

(3.6)

Page 8 of 20


3.2.1. A fixed point mapping associated with discrete problem (3.5) We consider the following mapping :

( )J 𝕋h : ℍ+ ⟶ 𝕍h ,

(3.7)

( ) W ↦ 𝕋h W = 𝜉h = 𝜉h1 , ..., 𝜉hJ , i

where 𝜉h ∈ 𝕍h is a solution of the following coercive system of QVI:

� � � � ⎧ bi 𝜉hi , vh − 𝜉hi ≥ f i + 𝜆.w i , vh − 𝜉hi , � � � � ⎪ ⎨ 𝜉hi ≤ rh k + 𝜉hi+1 , v ≤ rh k + 𝜉hi+1 ; ⎪ J+1 ⎩ 𝜉h = 𝜉h1 .

vh ∈ 𝕍h ; (3.8)

Theorem 3 Under the dmp and the preceding hypotheses and notations, the mapping 𝕋h is a con1 . Therefore, 𝕋h admits a unique fixed point which traction in ℍ+ with a rate of contraction 𝜌 = 𝛽Δt + 1 coincides with the solution of system (3.5). Proof It is very similar to that of the continuous case.

✷

3.3. A discrete iterative scheme 0

Starting from Uh = U0h, the solution of Equation (3.6), we define the sequence:

Uhn = 𝕋 Uhn−1 , n where Uh is

(3.9)

n = 1, ..., N,

a solution of the problem (3.5).

3.3.1. Geometrical convergence Proposition 3 Under the dmp and Theorem 3, we have : ‖U n − U ∞ ‖ ≤ ‖ h h ‖∞

(

1 𝛽Δt + 1

)n

‖ 0 ‖ ‖Uh − Uh∞ ‖ . ‖ ‖∞

(3.10)

where Uh∞ is the asymptotic solution of the problem of quasi-variational inequalities: find ( ) ( )J J,∞ Uh∞ = u1,∞ , ..., u ∈ 𝕍h such that h h � � � ⎧ i � i,∞ i,∞ ≥ f + 𝜆 ui,∞ , vh − ui,∞ , ⎪ b uh , vh − uh h h � � � � ⎪ i,∞ i+1,∞ i+1,∞ , vh ≤ rh k + uh ; ⎨ uh ≤ rh k + uh ⎪ J+1,∞ 1,∞ = uh . ⎪ uh ⎩

vh ∈ 𝕍h ;

Proof It is very similar to that of the continuous case.

(3.11)

✷

3.4. A monotonicity property i,n

(

Let uh = 𝜎h f

i,n

( ) ) = 𝜎h f̃ i,n ; k̃ ) the solution to (3.5). ; k (resp. ũ i,n h

Lemma 3 If f i,n ≥ f̃ i,n, and k ≤ k̃ then ui,n ≥ ũ i,n . h h

(3.12)

3.5. Lipschitz dependence with respect to the right-hand sides and parameter k Proposition 4 Under dmp and conditions of Lemma 3, we have ) ( ‖ | ‖ ‖ ‖ | max‖ui,n − ũ i,n ‖ ≤ C max |k − k̃ | + ‖f i,n − f̃ i,n ‖ , ‖∞ | ‖ ‖∞ 1≤i≤J | 1≤i≤J ‖

(3.13) Page 9 of 20


where C is a constant such that

a0 C ≥ 1.

(3.14)

Proof It is very similar to that of the continuous case.

✷

3.6. Characterization of the solution of problem (3.5) as the envelope of discrete subsolutions ( ) ( )J Definition 2 Zh = zh1 , ..., zhM ∈ 𝕍h is said to be a discrete subsolution for the system of quasi-variational inequalities (3.5) if � � � ⎧ i� i i i ⎪ b zh , 𝜑s ≤ f + 𝜆.zh , 𝜑s , � � ⎪ i i+1 ⎨ zh ≤ rh k + zh , 𝜑s ≥ 0; ⎪ J+1 1 ⎪ zh = zh . ⎩

� � ∀s, s = 1, ..., m h ;

(3.15)

Notation 4 Let 𝕏h be the set of such subsolutions. Theorem 4 Under the dmp, the solution of (3.5) is the least upper bound of the set 𝕏h .

3.7. The discrete regularity n

A discrete solution Uh of a system of quasi-variational inequalities is regular in the discrete sense if it satisfies: Theorem 5 There exists a constant C independent of k and h such that | i ( i,n )| |b u , 𝜑 | ≤ C ‖𝜑 ‖ 1 , ‖ s ‖L (Ω) | s | h | |

( ) s = 1, ..., m h .

(3.16)

{ } Moreover, there exists a family of right-hand sides gi,n (h)

‖ i,n ‖ ‖g(h) ‖ ≤ C ‖ ‖∞

h>0

( )J bounded in L∞ (Ω) such that

(3.17)

and ( ) ( ) bi ui,n , vh = gi,n , vh , vh ∈ 𝕍h . h (h)

(3.18)

Let ui,n be the corresponding continuous counterpart of (3.18), that is (h) ( ) ( ) i,n bi u , v = gi,n , v , v ∈ H01 (Ω), (h) (h)

(3.19)

then, there exists a constant C independent of k and h such that ‖ i,n ‖ ‖u ‖ ‖ (h) ‖ 2,P ≤ C, ‖ ‖W (Ω)

(3.20)

and ‖ i,n ‖ ‖u − ui,n ‖ ≤ Ch2 |log h|2 . | | ‖ (h) h ‖ ‖ ‖∞

Proof We adapt [.].

(3.21) ✷

Page 10 of 20


Remark 2 This new concept of “discrete regularity”, introduced in Berger and Falk (1977), CorteyDumont (1985) (see also Boulbrachene and Cortey-Dumont, 2009; Boulbrachene, 2015b), can be re‖ ‖ garded as the discrete counterpart of the lewy-Stampacchia regularity estimate ‖Ai u‖ ≤ C extend‖ ‖∞ ∞ 1 ed to the variational form through the L − L duality. It plays a major role in deriving the optimal 2,P error estimate as it permits to regularize the discrete obstacle “k + ui+1 (Ω) regular ones. h ” with W

4. Finite element error analysis This section is devoted to demonstrate that the proposed method is optimally accurate in L . We first introduce the following two auxiliary systems : ∞

4.1. Definition of two auxiliary sequences of elliptic variational inequalities 4.1.1. A discrete sequence ( { of}variational inequalities

̄n We define the sequence U h tional inequalities (VI):

1,n

n

n≥1

J,n

̄ = ū , ..., ū such that U h h h

)

solves the discrete system of varia-

� � � ⎧ i � i,n i,n i,n i,n−1 , vh − ū i,n , vh ∈ 𝕍h ; ⎪ b ū h , v − ū h ≥ f + 𝜆.u h � � � � ⎨ i,n i+1,n−1 i+1,n−1 , vh ≤ rh k + u , ⎪ ū h ≤ rh k + u ⎩ ( 1,n ) n J,n is the solution of the continuous problem (2.6). where U = u , ..., u

(4.1)

Proposition 5 There exists a constant C independent of h, △t and k such that 2 ‖ ‖ max‖ū i,n − ui,n ‖ ≤ Ch2 ||log h|| . (4.2) ‖∞ 1≤i≤J ‖ h ( ) ( ( )) Proof Since ū i,n = 𝜎h f i,n ; rh k + ui+1,n−1 is the approximation of ui,n = 𝜎 f i,n ; k + ui+1,n−1 . So, makh ✷ ing use of Cortey-Dumont (1985), we get the desired result.

4.1.2. A continuous sequence ( { }of variational inequalities n

̄ We define the sequence U (h) variational inequalities (VI):

n≥1

n (h)

̄ such that U

) = ū 1,n , ..., ū J,n solves the continuous system of (h) (h)

� � � ⎧ i � i,n i,n i,n−1 i,n 1 i,n ⎪ b ū (h) , v − ū (h) ≥ f + 𝜆.uh , v − ū (h) , v ∈ H0 (Ω); (4.3) ⎨ i,n i+1,n−1 i+1,n−1 ⎪ ū (h) ≤ k + u(h) , v ≤ k + u(h) , ⎩ ( ) ) ( 1,n M,n 1,n M,n n n is the solution of the discrete problem (3.5), and U h = u , ..., u is where Uh = uh , ..., uh () (h) (h) the solution of Equation (3.19).

Proposition 6 There exists a constant C independent of h, k and △t such that ‖ i,n ‖ i,n ‖ 2| |2 max‖ ‖ū − uh ‖ ≤ Ch |log h| . 1≤i≤J ‖ (h) ‖∞

(4.4)

Proof We adapt Boulbrachene (2015b).

✷

Lemma 4 (cf. Nochetto & Sharp, 1988) There exists a constant C independent of h, k and Δt such that ‖ ‖ max‖ui0 − uih0 ‖ ≤ C h2 ||log h||. ‖∞ 1≤i≤M ‖

(4.5)

4.1.3. Optimal L∞-error estimates n

n

Here, we shall estimate the error in the L –norm between the n th iterates U and Uh defined in (2.11) and (3.9), respectively. ∞

Page 11 of 20


Theorem 6 Under the previous hypotheses, there exists a constant C independent of h , k and △t such that ‖Un − Un ‖ ≤ Ch2 |log h|2 . ‖ | | h ‖∞

(4.6)

The proof is based on two Lemmas: ( ) Lemma 5 There exists a sequence of discrete subsolutions 𝜶 nh = 𝛼h1,n , ..., 𝛼hM,n such that ⎧ i,n i,n i = 1, ..., J; ⎪ 𝛼h ≤ uh , ⎪ ⎨ and 2 � ⎪ max� 𝛼 i,n − ui,n � ≤ Ch2 ��log h�� , ⎪ 1≤i≤J � � h �∞ ⎩

(4.7)

where the constant C is independent of h, k and △t. Proof For n = 1, we consider the discrete system of variational inequalities � � � ⎧ i � i,1 i,1 i,1 i,1 i ⎪ b ū h , vh − ū h ≥ f + 𝜆u0 , vh − ū h , � � � � ⎨ i,1 i+1 i+1 ⎪ ū h ≤ rh k + u0 , vh ≤ rh k + u0 , ⎩

v ∈ 𝕍h ;

Then, as ū 1h is solution to a discrete variational inequalities, it is also a subsolution, i.e. � � � ⎧ i � i,1 i,1 i ⎪ b ū h , 𝜑s ≤ f + 𝜆u0 , 𝜑s , � � ⎨ i,1 i+1 ⎪ ū h ≤ rh k + u0 , ⎩

∀𝜑s ;

or � � � ⎧ i � i,1 i,1 i i i ⎪ b ū h , 𝜑s ≤ f + 𝜆u0 − 𝜆u0h + 𝜆u0h , 𝜑s ; � � ⎨ i,1 i+1 ⎪ ū h ≤ rh k + u0 . ⎩

Then � � � ⎧ i � i,1 � i i,1 i � + 𝜆ui0h , 𝜑s ; ⎪ b ū h , 𝜑s ≤ f + 𝜆�u0 − u0h � � � � ��∞ � � ⎨ i,1 i+1 + rh k + ui+1 − rh k + ū i+1 ⎪ ū h ≤ rh k + u0 0 0h . ⎩

It follows � � � ⎧ i � i,1 � � i i,1 u0 − ui0h � + 𝜆ui0h , 𝜑s ; ⎪ b ū h , 𝜑s ≤ f + 𝜆� � � ∞ ⎨ 1 � i+1 ̄ i+1 � u − u0h � + ū i+1 ⎪ ū h ≤ k + � 0h . � 0 �∞ ⎩

Using (4.5), we have � � � ⎧ i � i,1 i,1 2 i ⎪ b ū h , 𝜑s ≤ f + C h ��log h�� + 𝜆u0,h , 𝜑s ; ⎨ i,1 2 i+1 ⎪ ū h ≤ k + C h ��log h�� + ū 0h . ⎩ Page 12 of 20


So, ū 1h (is a discrete subsolution for the) quasi-variational ( inequalities) whose solution is . Then, as ui,1 ; making use of Prop= 𝜎h f i,1 ; k + ū i+1 Ū hi,1 = 𝜎h f i,1 + C h2 ||log h||; k + C h2 ||log h|| + ū i+1 0h 0h h osition 4, we have ) ( ‖ | ‖ ‖ ‖ ̄ i,1 | ‖ ≤ C |k + C h2 ||log h|| − k| + ‖f i,1 + C h2 ||log h|| − f i,1 ‖ ‖Uh − ui,1 h ‖∞ ‖∞ | ‖ | ‖ ≤ C h2 ||log h|| + C h2 ||log h|| ≤ C h2 ||log h||.

Hence, making use of Theorem 4, we have ū i,1 ≤ Ū hi,1 ≤ ui,1 + C h2 ||log h||. h h

Putting 𝛼hi,1 = ū i,1 − C h2 ||log h||, h

we get 𝛼hi,1 ≤ ui,1 , h

and ‖ i,1 ‖ ‖ ‖ − C h2 ||log h|| − u1 ‖ ‖𝛼h − ui,1 ‖ = ‖ū i,1 ‖ ‖∞ ‖ h ‖∞ ‖ ̄ i,1 i,1 ‖ 2| ≤ ‖uh − u ‖ + C h |log h||. ‖ ‖∞

Using Proposition 5, we get 2 ‖ i,1 ‖ ‖𝛼h − ui,1 ‖ ≤ C h2 ||log h|| + C h2 ||log h|| ‖ ‖∞ 2 ≤ C h2 ||log h|| .

For n + 1, let us now assume that ⎧ ⎪ ⎨ ⎪ ⎩

𝛼hi,n ≤ ui,n , h and 2 � i,n � �𝛼h − ui,n � ≤ C h2 ��log h�� , � �∞

and we consider the system � � � ⎧ i � i,n+1 i,n+1 ≥ f i,n + 𝜆ui,n , vh − ū i,n+1 ; ⎪ b ū h , vh − ū h h � � � � ⎨ i,n+1 i+1,n i+1,n , vh ≤ rh k + u . ⎪ ū h ≤ rh k + u ⎩

Then � � � ⎧ i � i,n+1 i,n i,n ⎪ b ū h , 𝜑s ≤ f + 𝜆u , 𝜑s ; � � ⎨ i,n+1 i+1,n , ⎪ ū h ≤ rh k + u ⎩

or � � � ⎧ i � i,n+1 i,n i,n i,n i,n ⎪ b ū h , 𝜑s ≤ f + 𝜆u − 𝜆 ū h + 𝜆ū h , 𝜑s ; � � � � � � ⎨ i,n+1 i+1,n + rh k + ū i+1,n − rh k + ū i+1,n . ⎪ ū h ≤ rh k + u h h ⎩ Page 13 of 20


Then � � � ⎧ i � i,n+1 � i,n ̄ i,n � i,n u − uh � + 𝜆ū i,n , 𝜑 ; ⎪ b ū h , 𝜑s ≤ f + 𝜆� s h � �∞ ⎨ i,n+1 � i+1,n ̄ i+1,n � i+1,n ̄ ̄ u − uh � + uh . ⎪ uh ≤ k + � � �∞ ⎩

Using (4.2), we have � � � ⎧ i � i,n+1 2 i,n i,n 2 ⎪ b ū h , 𝜑s ≤ f + C h ��log h�� + 𝜆ū h , 𝜑s ; ⎨ i,n+1 2 i+1,n 2 ⎪ ū h ≤ k + C h ��log h�� + ū h . ⎩

So, ū hi,n+1 is a discrete subsolution for the quasi-variational inequalities whose solution is ( ) ( ) 2 2 . Then, as ui,n+1 , making use = 𝜎h f i,n ; k + ui+1,n Ū hi,n+1 = 𝜎h f i,n + C h2 ||log h|| ; k + C h2 ||log h|| + ū i+1,n h h h of Proposition 4, we have ) ( 2 2 ‖ | ‖ ‖ ‖ ̄ i,n+1 | ‖ ≤ C |k + C h2 ||log h|| − k| + ‖f i,n + C h2 ||log h|| − f i,n ‖ ‖Uh − ui,n+1 h ‖∞ | ‖ ‖∞ ‖ | 2 ≤ C h2 ||log h|| .

Hence, applying Theorem 4, we get 2 ū i,n+1 ≤ Ū hi,n+1 ≤ ui,n+1 + C h2 ||log h|| . h h

Putting 𝛼hi,n+1 = ū i,n+1 − C h2 ||log h|| , h 2

we get 𝛼hi,n+1 ≤ uhi,n+1 ,

and 2 ‖ i,n+1 ‖ ‖ ‖ − C h2 ||log h|| − un+1 ‖ ‖𝛼h − ui,n+1 ‖ = ‖ū i,n+1 ‖ ‖∞ ‖ h ‖∞ 2 ‖ ̄ i,n+1 i,n+1 ‖ 2| ≤ ‖uh − u ‖ + C h |log h|| . ‖ ‖∞

Using Proposition 5, we get 2 ‖ i,n+1 ‖ ‖𝛼h − ui,n+1 ‖ ≤ C h2 ||log h|| , ‖ ‖∞


✷

( ) Lemma 6 There exists a sequence of continuous subsolutions 𝛽 nh = 𝛽 1,n , ..., 𝛽 M,n such that () (h) (h) ⎧ i,n i,n i = 1, ..., J; ⎪ 𝛽(h) ≤ u , ⎪ ⎨ and � i,n � ⎪ �𝛽 − ui,n � ≤ Ch2 �log h�2 , ⎪ max � � � h � �∞ ⎩ 1≤i≤J � (h)

(4.8)

where the constant C is independent of h, k and △t. Proof For n = 1, we consider the system of variational inequalities

Page 14 of 20


� � � ⎧ i � i,1 i,1 i,1 i,1 i ⎪ b ū (h) , v − ū (h) ≥ f + 𝜆u0h , v − ū (h) , ⎨ i,1 i+1 i+1 ⎪ ū (h) ≤ k + u0(h) , v ≤ k + u0(h) , ⎩

v ∈ H01 (Ω);

Then, as ū i,1 is solution to a continuous variational inequalities, it is also a subsolution, i.e., (h) � � � ⎧ i � i,1 i,1 i ⎪ b ū (h) , v ≤ f + 𝜆u0h , v ; ⎨ i,1 i+1 ⎪ ū (h) ≤ k + u0(h) , ⎩

or � � � � ⎧ i i,1 i,1 i i i ⎪ b ū (h) , v ≤ f �+ 𝜆u0h −�𝜆u0�+ 𝜆u0 , v� ; ⎪ i,1 i+1 i+1 i+1 ⎨ ū (h) ≤ k + u0(h) + k + u0h − k + u0h � � � ⎪ � i+1 i+1 ⎪ + k + ū 0(h) − k + ū 0(h) . ⎩

Then � � � � ⎧ bi ū i,1 , v ≤ f i,1 + 𝜆�ui − ui � + 𝜆ui , v ; � � 0 0h 0 ⎪ � �∞ (h) � � � � � i+1 � i+1 ⎨ i,1 i+1 i+1 � i+1 � �ū − − u + u ⎪ ū (h) ≤ � � 0(h) u0h � + k + ū 0(h) . � 0(h) 0h � �∞ �∞ � � ⎩

Using (4.5), we have � � � ⎧ i � i,1 i,1 2� i � ⎪ b ū (h) , v ≤ f + C h �log h� + 𝜆u0 , v ; ⎨ i,1 2 i+1 ⎪ ū (h) ≤ k + C h ��log h�� + ū 0(h) . ⎩

So, ū 1h is a continuous subsolution for the variational inequalities whose solution is ( )( ( ) ) 2 ; making use of Ū i,1 = 𝜎 f i,1 + C h2 ||log h||; k + C h2 ||log h|| + ū i+1 . Then, as ui,1 = 𝜎 f i,1 ; k + ui+1 0 0(h) (h) Proposition 2, we have ) ( ‖ i,1 ‖ ‖ ‖Ū − ui,1 ‖ ≤ C ||k + C h2 |log h| − k|| + ‖ ‖f i,1 + C h2 ||log h|| − f i,1 ‖ | | ‖ ‖ (h) ‖∞ | ‖ | ‖∞ ‖ 2| 2| | | ≤ C h |log h| + C h |log h| ≤ C h2 ||log h||.

Hence, making use of Theorem 2, we have ū i,1 ≤ Ū i,1 ≤ ui,1 + C h2 ||log h||. (h) (h)

Putting 𝛽 i,1 = ū i,1 − C h2 ||log h||, (h) (h)

we get 𝛽 i,1 ≤ ui,1 , (h)

Page 15 of 20


and ‖ ‖ ‖ i,1 ‖ ‖𝛽 − ui,1 ‖ = ‖ū i,1 − C h2 |log h| − u1 ‖ ‖ | | ‖ (h) h‖ h ‖ ‖∞ ‖∞ ‖ (h) ‖ ‖ ‖ i,1 i,1 2 | | ‖ ≤‖ ‖ū (h) − uh ‖ + C h |log h|. ‖∞ ‖

Using Proposition 6, we get ‖ i,1 ‖ ‖𝛽 − ui,1 ‖ ≤ C h2 |log h|2 + C h2 |log h| | | | | ‖ (h) h ‖ ‖ ‖∞ 2 2| ≤ C h |log h|| .

For n + 1, let us now assume that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

𝛽 i,n ≤ ui,n ; (h) and � i,n � �𝛽 − ui,n � ≤ C h2 �log h�2 , � � � (h) h � � �∞

and consider the system � � � ⎧ i � i,n+1 i,n+1 i,n i,n+1 i,n ⎪ b ū (h) , v − ū (h) ≥ f + 𝜆uh , v − ū (h) ; ⎨ i,n+1 i+1,n i+1,n ⎪ ū (h) ≤ k + u(h) , v ≤ k + u(h) . ⎩

Then � ⎧ i � i,n+1 � � i,n i,n ⎪ b ū (h) , v ≤ f + 𝜆uh , v ; ⎨ i,n+1 i+1,n ⎪ ū h ≤ k + u(h) , ⎩

or � � � � ⎧ i ̄ i,n+1 i,n i,n i,n i,n ̄ ̄ u b ≤ f + 𝜆u − 𝜆 u ; , v + 𝜆 u , v ⎪ (h) (h) � (h) � � � � h ⎪ i,n+1 � i+1,n i+1,n i+1,n + k + uh ⎨ ū (h) ≤ k + u(h) − k + uh � � � ⎪ � i+1,n i+1,n ⎪ + k + ū (h) − k + ū (h) . ⎩

Then � � � � � i,n � ⎧ i ̄ i,n+1 i,n �u − ū i,n � + 𝜆ū i,n , v ; u b ≤ f + 𝜆 , v � h ⎪ (h) (h) � (h) � �∞ � � ⎨ � i+1,n � � i+1,n � i,n+1 i+1,n i+1,n i+1,n � � � ⎪ ū ≤� �uh − ū (h) � + �uh − u(h) � + k + ū (h) . ⎩ (h) � �∞ � �∞

Using (4.4), we have � ⎧ i � i,n+1 � � i,n i,n 2� �2 ⎪ b ū (h) , v ≤ f + C h �log h� + 𝜆ū (h) , v ; � ⎨ i,n+1 � 2 � 2 i+1,n ⎪ ū (h) ≤ C h �log h�� + k + ū (h) . ⎩

is a continuous subsolution for the quasi-variational inequalities whose solution is So, ū n+1 (h) ( ( ) ) ̄Ui,n+1 = 𝜎 f i,n + C h2 |log h|2 ; k + C h2 |log h|2 + ū i+1,n . Then, as ui,n+1 = 𝜎 f i,n ; k + ui+1,n ; making use of | | | | h (h) (h) Proposition 2, we have

Page 16 of 20


) ( 2 2 ‖ | ‖ ‖ ‖ ̄ i,n+1 | ‖ ≤ C |k + C h2 ||log h|| − k| + ‖f i,n + C h2 ||log h|| − f i,n ‖ ‖Uh − ui,n+1 h ‖∞ | ‖ ‖∞ ‖ | 2 ≤ C h2 ||log h|| .

Hence, applying Theorem 2, we get 2 ū i,n ≤ Ū i,n+1 ≤ ui,n+1 + C h2 ||log h|| . (h) (h)

Putting 2 𝛽 i,n+1 = ū i,n+1 − C h2 ||log h|| , (h) (h)

we get 𝛽 i,n+1 ≤ ui,n+1 (h)

and ‖ ‖ i,n+1 ‖ i,n+1 ‖ i,n+1 ‖ 2| n+1 ‖ ‖ |2 ‖𝛽 ‖ (h) − uh ‖ = ‖ū (h) − C h |log h| − uh ‖ ‖∞ ‖∞ ‖ ‖ ‖ ‖ i,n+1 i,n+1 2 | |2 ‖ ≤‖ ‖ū (h) − uh ‖ + C h |log h| . ‖∞ ‖

Using Proposition 6, we get ‖ i,n+1 ‖ i,n+1 ‖ 2| |2 ‖𝛽 ‖ (h) − uh ‖ ≤ C h |log h| , ‖ ‖∞


✷

We are now in a position to prove the Theorem 6. Proof Using (4.7), we have ui,n ≤ 𝛼hi,n + C h2 ||log h||

2

≤ ui,n + C h2 ||log h|| h

2

thus 2 ui,n − ui,n ≤ C h2 ||log h|| h

and using (4.8), we have 2 ui,n ≤ 𝛽 i,n + C h2 ||log h|| h (h) 2 ≤ ui,n + C h2 ||log h|| ,

thus, we get 2 ui,n − ui,n ≤ C h2 ||log h|| . h

Therefore, 2 ‖ i,n ‖ ‖u − ui,n ‖ ≤ C h2 ||log h|| , h ‖∞ ‖


✷

Page 17 of 20


5. L∞-Asymptotic behavior for a finite element approximation 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 quasi-variational inequalities. Now, we evaluate the variation in L -norm between uh (T, .) the discrete solution calculated at the ∞ moment T = n Δt and u the solution of system (2.13). ∞

Theorem 7 (The main result) Under Propositions 1 and 3, and Theorem 6, the following inequality holds: ( ( )N ) 1 2 ‖ i i,∞ ‖ 2| | max‖uh (T, .) − u (.)‖ ≤ C h |log h| + . (5.1) ‖∞ 1≤i≤M ‖ 𝛽Δt + 1 Proof We have ] [ uih = uih (t, .) for allt ∈ (n − 1).Δt, n.Δt ,

thus ‖ i ‖ ‖ ‖ − ui,∞ ‖ ‖uh (T, .) − ui,∞ (.)‖ = ‖ui,N ‖ ‖∞ ‖ h ‖∞ ‖ ‖ ‖ i,∞ ‖ − ui,∞ ‖ . − u ≤ ‖ui,N ‖ + ‖ui,∞ h ‖∞ ‖ h ‖∞ ‖ h

Indeed, combining estimates (2.12), (3.10), and (4.6), we get ‖ i ‖ ‖ ‖ ‖ ‖ − ui,∞ − ui,∞ ‖ ‖ + ‖ui,∞ ‖uh (T, .) − ui,∞ ‖ ≤ ‖ui,N h ‖∞ ‖∞ ‖ h ‖ h ‖∞ ‖ ‖ ‖ ‖ i,∞ ‖ i,N ‖ i,∞ ‖ − ui,∞ ‖ + ‖ui,N u − u + − u ≤ ‖ui,N ‖ ‖ ‖ h ‖∞ h ‖∞ ‖ h ‖∞ ‖ h ‖ h ‖ i,N ‖ i,∞ ‖ i,N i,∞ ‖ i,N ‖ i,N ‖ ≤ 2‖uh − uh ‖ + ‖u − u ‖ + ‖u − uh ‖ . ‖ ‖∞ ‖ ‖∞ ‖ ‖∞

Using Propositions 1 and 3, we have ‖ i,∞ ‖ ‖u − ui,N ‖ ≤ ‖ ‖∞

(

1 𝛽Δt + 1

)N

‖ i,∞ ‖ ‖u − ui0 ‖ , ‖ ‖∞

and for the discrete case ‖ i,∞ ‖ ‖uh − ui,N ‖ ≤ h ‖∞ ‖

(

1 𝛽Δt + 1

)N

‖ i,∞ ‖ ‖uh − uih0 ‖ , ‖ ‖∞

Applying the previous results of Propositions 1, 3 and Theorem 6 we get )N )N ( ( 1 1 ‖ i,∞ ‖ i,∞ ‖ ‖ ‖ i ‖ ‖uh − uih0 ‖ + ‖u − ui0 ‖ ‖uh (T, .) − ui,∞ ‖ ≤ 2 ‖∞ ‖∞ ‖∞ ‖ 𝛽Δt + 1 ‖ 𝛽Δt + 1 ‖ + Ch2 ||log h||

2

Then, the following result can be deduced: ( ( )N ) 1 2 ‖ i i,∞ ‖ 2| | , ‖uh (T, .) − u ‖ ≤ C h |log h| + ‖ ‖∞ 𝛽Δt + 1 which completes the proof.

✷ (

)N

1 tends to 0 when 𝛽Δt + 1 N approaches to +∞. Therefore, the convergence order for the noncoercive elliptic system of

Corollary 1 It can be seen that in the previous estimate (5.1),

quasi-variational inequalities related to stochastic control problems is 2 ‖ ‖ max‖ui,∞ − ui,∞ ‖ ≤ Ch2 ||log h|| . ‖∞ 1≤i≤M ‖ h

(5.2) Page 18 of 20


Acknowledgements The second author gratefully acknowledge Qassim University in Kingdom of Saudi Arabia and all authors would like to thank the anonymous referees for their careful reading and for relevant remarks/suggestions which helped them to improve the paper. Funding The authors received no direct funding for this research. Author details Mohamed Amine Bencheikh Le Hocine1,2 E-mail: [email protected] Salah Boulaaras3,4 E-mail: [email protected] Mohamed Haiour2 E-mail: [email protected] 1 Tamanghesset University Center, B.P. 10034, Sersouf, Tamanghesset 11000, Algeria. 2 LANOS Laboratory, Faculty of Sciences, Department of Mathematics, Badji Mokhtar University, B.P. 12, Annaba 23000, Algeria. 3 Department of Mathematics, College of Sciences and Arts, Al-Ras, Qassim University, Buraydah, Kingdom of Saudi Arabia. 4 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria. Citation information Cite this article as: On finite element approximation of system of parabolic quasi-variational inequalities related to stochastic control problems, Mohamed Amine Bencheikh Le Hocine, Salah Boulaaras & Mohamed Haiour, Cogent Mathematics (2016), 3: 1251386. References Achdou, A., Hecht, F., & Pommier, D. (2008). A posteriori error estimates for parabolic variational inequalities. Journal of Scientific Computing, 37, 336–366. Alfredo, F. (1987). L∞-error estimate for an approximation of a parabolic variational inequality. Numerische Mathematik, 50, 557–565. Bensoussan, A., & Lions, J. L. (1973). ContrÔle impulsionnel et inéquations quasi-variationnelles d’évolution. Note CRAS Paris, 276, 1333–1338. Bensoussan, A., & Lions, J. L. (1978). Applications des inéquations variationnelles en controle stochastique. Paris: Dunod. Bencheikh Le Hocine, M. A., Boulaaras, S., & Haiour, M. (2016). An optimal L∞-error estimate for an approximation of a parabolic variational inequality. Numerical Functional Analysis and Optimization, 37, 1–18. Bencheikh Le Hocine, M. A., & Haiour, M. (2013). Algorithmic approach for the aysmptotic behavior of a system of parabolic quasi variational inequalities. Applied Mathematics Sciences, 7, 909–921. Berger, A., & Falk, R. (1977). An error estimate for the truncation method for the solution of parabolic obstacle variational inequalities. Mathematics of Computation., 31, 619–628.

Boulaaras, S., Bencheikh Le Hocine, M. E. D. A., & Haiour, M. (2014). The finite element approximation in a system of parabolic quasi-variational inequalities related to management of energy production with mixed boundary condition. Computational Mathematics and Modeling, 25, 530–543. Boulaaras, S., & Haiour, M. (2014). The theta time scheme combined with a finite element spatial approximation in the evolutionary Hamilton-Jacobi-Bellman equation with linear source terms. Computational Mathematics and Modeling, 25, 423–438. Boulbrachene, M. (2014). Optimal L∞-error estimate for a system of elliptic quasi-variational inequalities with noncoercive operators. Advances in Applied Mathematics (Springer Proceedings in Mathematics & Statistics), 87, 89–96. Boulbrachene, M. (2015a). On the finite element approximation of variational inequalities with noncoercive operators. Numerical Functional Analysis and Optimization, 36, 1107–1121. Boulbrachene, M. (2015b). On the finite element approximation of the impulse control quasivariational inequality. In A. H. Siddiqi, P. Manchanda, & R. Bhardwaj (Eds.), Mathematical Models, Methods and Applications. Industrial and Applied Mathematics (pp. 107–126). Singapore: Springer. Boulbrachene, M., & Cortey-Dumont, P. (2009). Optimal L∞-error estimate of a finite element method for Hamilton—Jacobi—Bellman equations. Numerical Functional Analysis and Optimization, 30, 421–435. Boulbrachene, M., Haiour, M., & Chentouf, B. (2002). On a noncoercive system of quasi-variational inequalities related to stochastic. Journal of Inequalities in Pure and Applied Mathematics, 3, 1–14. Cortey-Dumont, P. (1987). Sur l’analyse numerique des equations de Hamilton-Jacobi-Bellman. Mathematical Methods in the Applied Sciences, 9, 198–209. Cortey-Dumont, P. (1985). On the finite element approximation in the L∞-norm of variational inequalities with nonlinear operators. Numerische Mathematik, 47, 45–57. Cortey-Dumont, P. (1985). Sur les inéquations variationnelles opérateur non coercif. Modelisation Mathematique et Analyse Numerique, 19, 195–212. Cortey-Dumont, P. (1983). Contribution a l’approximation des inequations variationnelles en norme L∞. Comptes Rendus de l’Acadmie des Sciences - Series I - Mathematics, 17, 753–756. Diaz, I., & Defonso, J. (1985). On a Fully non-linear parabolic equation and the asymptotic behavior of its solution. Journal of Mathematical Analysis and Applications, 15, 144–168. Evans, L. C., & Friedman, A. (1979). Optimal stochastic switching and the Dirichlet Problem for the Bellman equations. Transactions of the American Mathematical Society, 253, 365–389. Lions, P. L., & Menaldi, J. L. (1979). Optimal control of stochastic integrals and Hamilton Jacobi Bellman equations (Part I). SIAM Control and Optimization, 20, 58–81. Nochetto, R. H. (1988). Sharp L∞-error estimates for semilinear elliptic problems with free boundaries. Numerische Mathematik, 54, 243–255. Scarpini, F., & Vivaldi, M. A. (1977). Evaluation de l’erreur d’approximation pour une inéquation parabolique relative aux convexes d épendant du temps. Applied Mathematics & Optimization, 4, 121–138.

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