On finite element approximation of system of parabolic quasi-variational inequalities related to stochastic control problems

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.

In the current paper, we shall employ the concepts of subsolutions and discrete regularity Boulbrachene, 2014Boulbrachene, , 2015aBoulbrachene, , 2015bCortey-Dumont, 1987. 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): 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. (1.9)

The time discretization
We discretize the problem (1.1) or (1.7) with respect to time by using the semi-implicit scheme.
Therefore, we search a sequence of elements u i,n ∈ H 1 0 (Ω), 1 ≤ i ≤ J, which approaches u i t n , t n = k Δt, with initial data u i,0 = u i,0 .
Thus, we have for n = 1, ..., N where By adding , v − u i,n to both parties of the inequalities (2.1), we get The bilinear form a(., .), being noncoercive in H 1 0 (Ω), there exist two constants > 0 and > 0 such that:

Notation 1
We denote by u i,n = f i,n ; k + u i+1,n the solution of problem (2.6).
0 be the solution of the following continuous equation: The existence and uniqueness of a continuous solution is obtained by means of Banach's fixed point theorem.

A fixed point mapping associated with continuous system (2.6)
. We introduce the following mapping: where solves the following coercive system of QVI: Theorem 1 Under the preceding hypotheses and notations, the mapping is a contraction in ℍ + with a contraction constant = 1 Δt + 1 . Therefore, admits a unique fixed point which coincides with the solution of problem (2.6).

A continuous iterative scheme
Starting from U 0 = U 0 , the solution of Equation (2.8), we define the sequence: where U n is a solution of the problem (2.6).

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: (2.9) where U ∞ is the asymptotic solution of the problem of quasi-variational inequalities: find Proof Under Theorem 1, we have for n = 1 Now, we assume that then Thus, which completes the proof. ✷ 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.

A monotonicity property
Let k and k be two parameters, f 1,n , ..., f J,n and f 1,n , ...,f J,n be two families of right-hand sides.

Lipschitz dependence with respect to the right-hand sides and the parameter k
Proposition 2 Under conditions of Lemma 1, we have where C is a constant such that Proof Let Then, from (2.16) it is easy to see that and So, due to Lemma 1 it follows that hence Interchanging the role of f i,n and f i,n , k and k we also get Then which completes the proof. ✷

Characterization of the solution of system (2.6) as the envelope of continuous subsolutions
Definition 1 Z = z 1 , ..., z J ∈ H 1 0 (Ω) J is said to be a continuous subsolution for the system of quasi-variational inequalities (2.6) if Notation 2 Let 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 .

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.

Spatial discretization
Let M s , 1 ≤ s ≤ m h denote the vertex of the triangulation h , and let l , 1 ≤ l ≤ m h , denote the functions of h which satisfies: So that the function l from a basis of h . ∀v h ∈ L 2 0, T; H 1 0 (Ω) ∩ C 0, T; H 1 0 Ω represents the interpolate of v over h .

Existence and uniqueness
The discrete problem of PQVI consists of seeking U h = u 1 h , ..., u J h ∈ h M such that or equivalently,

Notation 3
We denote by u i,n h = h f i,n ; r h k + u i+1,n h the solution of system (3.5).
Let U 0 h = U 0h = u 1 0h , ..., u M 0h be the solution of the following discrete equation:

A fixed point mapping associated with discrete problem (3.5)
We consider the following mapping : where i h ∈ h is a solution of the following coercive system of QVI: Theorem 3 Under the dmp and the preceding hypotheses and notations, the mapping h is a contraction in ℍ + with a rate of contraction = 1 Δt + 1 . Therefore, h admits a unique fixed point which coincides with the solution of system (3.5).
Proof It is very similar to that of the continuous case. ✷

A discrete iterative scheme
Starting from U 0 h = U 0h , the solution of Equation (3.6), we define the sequence: where U n h is a solution of the problem (3.5).

Geometrical convergence
Proposition 3 Under the dmp and Theorem 3, we have : where U ∞ h is the asymptotic solution of the problem of quasi-variational inequalities: find Proof It is very similar to that of the continuous case. ✷

A monotonicity property
Let u i,n h = h f i,n ; k (resp. ũ i,n h = h f i,n ;k ) the solution to (3.5).
Lemma 3 If f i,n ≥f i,n , and k ≤k then

Lipschitz dependence with respect to the right-hand sides and parameter k
Proposition 4 Under dmp and conditions of Lemma 3, we have (3.11) where C is a constant such that Proof It is very similar to that of the continuous case. ✷

The discrete regularity
A discrete solution U n h 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
Moreover, there exists a family of right-hand sides g i,n (h) h>0 bounded in L ∞ (Ω) J such that and Let u i,n (h) be the corresponding continuous counterpart of (3.18), that is then, there exists a constant C independent of k and h such that and Proof We adapt [.]. ✷ (3.14) a 0 C ≥ 1.
Remark 2 This new concept of "discrete regularity", introduced in Berger and Falk (1977),  (see also Boulbrachene and Cortey-Dumont, 2009;Boulbrachene, 2015b), can be regarded as the discrete counterpart of the lewy-Stampacchia regularity estimate ‖ ‖ ‖ A i u ‖ ‖ ‖∞ ≤ C extended to the variational form through the L ∞ − L 1 duality. It plays a major role in deriving the optimal error estimate as it permits to regularize the discrete obstacle "k + u i+1 h " with W 2,P (Ω) regular ones.

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 :

A discrete sequence of variational inequalities
We define the sequence Ū n h n≥1 such that Ū n h = ū 1,n h , ...,ū J,n h solves the discrete system of variational inequalities (VI): where U n = u 1,n , ..., u J,n is the solution of the continuous problem (2.6).
Proposition 5 There exists a constant C independent of h, △t and k such that Proof Since ū i,n h = h f i,n ; r h k + u i+1,n−1 is the approximation of u i,n = f i,n ; k + u i+1,n−1 . So, making use of , we get the desired result. ✷

A continuous sequence of variational inequalities
We define the sequence Ū n (h) n≥1 such that Ū n (h) = ū 1,n Proposition 6 There exists a constant C independent of h, k and △t such that Proof We adapt Boulbrachene (2015b). ✷ Lemma 4 (cf. Nochetto & Sharp, 1988) There exists a constant C independent of h, k and Δt such that 4.1.3. Optimal L ∞ -error estimates Here, we shall estimate the error in the L ∞ -norm between the n th iterates U n and U n h defined in (2.11) and (3.9), respectively.

Theorem 6 Under the previous hypotheses, there exists a constant C independent of h , k and △t such that
The proof is based on two Lemmas: Lemma 5 There exists a sequence of discrete subsolutions n h = 1,n h , ..., M,n h such that where the constant C is independent of h, k and △t.
Proof For n = 1, we consider the discrete system of variational inequalities Then, as ū 1 h is solution to a discrete variational inequalities, it is also a subsolution, i.e.
or Then It follows Using (4.5), we have (4.7)

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 u h (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: Proof We have thus Indeed, combining estimates (2.12), (3.10), and (4.6), we get Using Propositions 1 and 3, we have and for the discrete case Applying the previous results of Propositions 1, 3 and Theorem 6 we get Then, the following result can be deduced: which completes the proof. ✷ Corollary 1 It can be seen that in the previous estimate (5.1), 1 Δt + 1 N tends to 0 when N approaches to +∞. Therefore, the convergence order for the noncoercive elliptic system of quasi-variational inequalities related to stochastic control problems is