# L^{∞}-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms

### Salah Boulaaras

,### Med Amine Bencheikh Le Hocine

,### Mohamed Haiour

Journal of Applied Mathematics and Computational Mechanics |
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L^{∞}-ERROR ESTIMATES OF FINITE
ELEMENT METHODS
WITH EULER TIME DISCRETIZATION SCHEME
FOR AN EVOLUTIONARY HJB EQUATIONS
WITH NONLINEAR SOURCE TERMS

Salah Boulaaras^{ }^{1,2}, MedAmine Bencheikh Le Hocine^{ }^{3,4}, Mohamed Haiour^{ }^{3}

^{1} Department
of Mathematics, College of Science and Arts, Ar-Ras, Qassim University

Kingdom Of Saudi Arabia

^{2} Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO),

University of Oran 1, Ahmed Benbella. Algeria, Tel +966559618327.

^{3} Department of Mathematics, Faculty of Science, University of
Annaba, Box. 12

Annaba 23000. Algeria

^{4} Tamanghesset
University Center, Box. 10034, Sersouf, Tamanghesset 11000, Algeria.

S.Boularas@qu.edu.sa or saleh_boulaares@yahoo.fr, kawlamine@gmal.com,
haiourm@yahoo.fr

Received: 16 May 2016; accepted:
3 January 2017

**Abstract.** The
main purpose of this paper is to analyze the convergence of the proposed algorithm
[5] of the finite element methods coupled with a Euler discretization scheme.
Also, an optimal error estimate with an asymptotic behavior in uniform norm are
given for an evolutionary nonlinear Hamilton Jacobi Bellman (HJB) equation with
respect to the Dirichlet boundary conditions.

*MSC 2010:** 65M80, 35L20*

*Keywords: **QVIs, Finite elements,
Theta Scheme Fixed point, HJB equations, Geometric Convergence .*

1. Introduction

In this paper, we extend our work [5] and continue to analyze the convergence of the proposed algorithm of the finite element methods coupled with a Euler discretization scheme. In addition, an optimal error estimate with an asymptotic behavior in uniform norm are given, for the following evolutionary nonlinear HJB equation

(1)

where is a bounded open domain of , with boundary sufficiently smooth and set in , with the are given smooth positive functions, and the are second-order, uniformly elliptic operators defined over

(2)

and whose coefficients are sufficiently smooth coefficients and satisfy the following conditions

(3)

with

(4)

and the bilinear forms associated with , for

(5)

is a regular function satisfying

(6)

We shall also need the following norm

Let be the scalar product in

In (cf. [3]), we applied a new time-space discretization using the semi-implicit time scheme combined with a finite element approximation, we found (1) can be transformed into the following full-discrete HJB equation

(7)

where , such that defined on (2), , respectively

In [3-5] we proved the theorem of the geometrical convergence and the existence and uniqueness of the solution of both the continuous and the discrete HJB equation of the stationary case using Bensoussan's algorithm. Also, in (cf. [3,4]) the system of parabolic quasi variational inequalities (PQVIs) can be transformed into a system of the following full-discrete system of strongly coercive elliptic quasi variational inequalities (QVIs): find solution of

(8)

with

(9)

where the discrete spaces of finite element given by

(10)

where is the usual interpolation operator defined by

(11)

and denote the set of all those elements, is the mesh size and
it is regular and quasi-uniform. Moreover, the usual basis of affine functions
defined by ,
where is a sum of
triangulation mesh and be the M-matrices [8] with generic entries* *

(12)

and is an operator defined by

(13)

with and is a continuous operator from into itself satisfying the following assumptions

The class of the system of QVIs with coercive bilinear form includes at least two well-known important problems: the system of variational inequality of feedback obstacle (VIs) (when and ), and the system of quasi-variational inequality related to management of energy production (when is identically equal to , , (cf. [1]).

The evolutionary HJB equations (1) have many applications in science, engineering and economics; see for example [1] and references therein. They can arise in solving optimal control problems by dynamic programming techniques. Many nonlinear option pricing problems can also be formulated as optimal control problems, leading to HJB equations.

In the last few decades, many numerical schemes have been proposed for solving the stationary HJB equations; see for example [7-9] and references therein. Lions and Mercier [14] presented two iterative algorithms for solving HJB equations. At each iteration, a linear complementary subproblem or a linear equation system subproblem is solved. Boulbrachene and Haiour [6], by means of a subsolution method, conducted a finite element approximation study for the first time, for the stationary of the problem (1) and by using Bensoussan--Lions algorithm [1], a quasi-optimal error estimate in the - norm has been derived according the following result

In [4], exploiting the above arguments, where we analyzed the theta time scheme combined with a finite element spatial approximation for an evolutionary HJB equation with linear source terms and we derived the following error estimate

with a constant independent of both (step of the space discretization) and (step of the time discretization), where the discrete solution calculated at the moment-end and the asymptotic continuous solution with respect to the right hand side condition. In addition, we extended the above result [3] to nonlinear case but with the new generalized space-time discretization stands using the theta scheme and we obtained the following result

where is the rate of contraction of the nonlinear source term satisfying

(14)

with

(15)

In this paper, an -error estimate is established combining the geometric convergence of discrete iterative schemes using the known -error estimates for stationary and evolutionary free boundary problems (cf., e.g., [4,6]) which play a major role in the finite element error analysis section. Finally the asymptotic behavior in uniform norm is deduced which investigated the evolutionary free boundary problem similar to that in [3].

The structure of this paper is as follows. In Section 2 and 3, we consider the discrete system of quasi-variational inequalities, discretize the iterative scheme by the standard finite element method combined with a theta scheme and an algorithm iterative discrete scheme is introduced. Then its geometric convergence is proved with respect to-stability of the solution and the right-hand side and its characterization as the least upper bound of the subsolutions set (see also [5,6]). It is worth mentioning that this approach is entirely different from the one developed for the evolutionary problem. Also, it is used for the first time for a system of stationary QVIs. In Section 4, a fundamental lemma and given optimal error estimates with an asymptotic behavior in uniform norm are proved for the presented problem. Finally, we make some comments on the approach and the results presented in this paper.

2. The discrete coercive system of QVIs

**Definition 1**: is said to
be a subsolution for the system of QVIs (8) if

(16)

**Notation 1**: Let be the set of
discrete subsolutions. Then, we have the
following theorem.

**Theorem 1: [5]*** *Under the
discrete maximum principle, the solution of the
system of QVI (8) is the maximum element of .

2.1. Existence and uniqueness

2.1.1. A fixed point mapping associated with the system of QVIs

In [3], we have proved the existence and uniqueness of the discrete QVIs (8) using the algorithm based on semi-implicit time scheme combined with a finite element method, which has already been used in our previous research regarding the evolutionary free boundary problems (see [4, 5]).

For that, let us first introduce the initial vector where for is solution of

(17)

where

Let , where denotes the positive cone of .

Now, we consider the mapping

(18)

where is solution of the following problem

(19)

where

2.2. A discrete iterative scheme

Starting from solution of (17), we define

(20)

and

(21)

where is the subsolution of the problem (8).

**Theorem 2: [5]*** *The
sequences well
defined in and
converge to the unique solution of system of inequalities (8).

where

(22)

2.3. Regularity of sequences of HJB (21)

**Theorem 3: ** **(Lewy Stampacchia inequality)[1]** Let be an elliptic
operator defined in (2) and the
solution of an elliptic variational inequalities (VIs) with a simple obstacle in and the right hand
side such that in
the sense of , where Then

(23)

(24)

and

(25)

**Lemma 1: ** For , we have

(26)

where is a subsolution of the problem (8).

** Proof: ** It
is clear that

is the solution of (8) with the obstacle and the right hand side and

Since

then, we have

where

Using Lewy Stampacchia inequality, we get

where with .

Now, we assume that

then verify

and we have in , then

3. Geometrical convergence of the discrete algorithm

**Lemma 2:[5] *** *For , where is
a positive constant
defined in (13), then we have

(27)

**Proposition 1: [5] *** *Let
such that

(28)

then, we have

(29)

**Proposition 2**: Under the
assumptions and previous notations, we have

(30)

where is the subsolution of (8), and is an asymptotic semi-discrete solution of (1) using the standard finite element method.

**Proof **: Using
Theorem 2, we have

then

Using Proposition 1 with we get

or

Using Proposition 1 again with , we get

i.e.

and by induction

Under the third step of the proof of Theorem 3 in [5], we deduce that

4. Optimal error estimates and asymptotic behavior

Before discussing the results, it is interesting to introduce the result of the following problems

(31)

where solution of (17) and is the subsolution of (8).

**Theorem 4:** For all and is a constant
independent with ,
we have the following estimate

(32)

where is the subsolution of a semi-discrete problem in time using the semi-implicit scheme.

**Proof:** The proof is similar to
that in [8].

The following lemma will play a crucial role in obtaining the approximation error:

** Lemma 3:**
For all and independent by we have the following
estimate

(33)

**Proof **: By induction, we
have

then

Since is Lipschitz, thus

Assume that

then, we have

Using the induction assumption, we get

5. -optimal error estimate

**Theorem 5**:
For all and independent by , we have the following
estimate

(34)

**Proof :** We have

From the initial data in (1), we have and then, it can be used the following standard error estimate [8,9] which investigated the stationary case

(35)

Using the estimates (30), (32) and (35), thus

Setting

then

Therefore, it can be deduced

**Proposition 3: [3]** Under the assumption (14), we have for all the following
estimates

(36)

Now we evaluate the variation in - norm between the discrete solution calculated at the moment and the asymptotic continuous solution of (1).

**Theorem 6**: Under the results of
Proposition 3 and Theorem 5, we have for

(37)

** Proof**:
Using Theorem 5 and Proposition 3, it can be easily obtained

which completes the proof.

6. Conclusions

In this paper, the regularity and convergence of the presented algorithm sequences of the finite element methods coupled with the Euler time discretization scheme are analyzed. Also, an optimal error estimate with asymptotic behavior in a uniform norm are given for an evolutionary HJB equation with respect to the same proposed boundary conditions in [4]. A next paper will propose a decomposition methods for solving these problems. The convergence of the new scheme will be established and the numerical example will be shown to prove that the new presented scheme is efficient.

Acknowledgement

The first author gratefully acknowledge Qassim University in Kingdom of Saudi Arabia and this presented work in memory of his father (1910-1999) Mr. Mahmoud ben Mouha Boulaaras. All authors of this paper would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions which helped him to improve the paper.

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