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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@article{Boulaaras_2017, doi = {10.17512/jamcm.2017.1.02}, url = {https://doi.org/10.17512/jamcm.2017.1.02}, year = 2017, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {16}, number = {1}, pages = {19--31}, author = {Salah Boulaaras and Med Amine Bencheikh Le Hocine and Mohamed Haiour}, title = {L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms}, journal = {Journal of Applied Mathematics and Computational Mechanics} }
TY - JOUR DO - 10.17512/jamcm.2017.1.02 UR - https://doi.org/10.17512/jamcm.2017.1.02 TI - L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms T2 - Journal of Applied Mathematics and Computational Mechanics JA - J Appl Math Comput Mech AU - Boulaaras, Salah AU - Hocine, Med Amine Bencheikh Le AU - Haiour, Mohamed PY - 2017 PB - The Publishing Office of Czestochowa University of Technology SP - 19 EP - 31 IS - 1 VL - 16 SN - 2299-9965 SN - 2353-0588 ER -
Boulaaras, S., Hocine, M., & Haiour, M. (2017). L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms. Journal of Applied Mathematics and Computational Mechanics, 16(1), 19-31. doi:10.17512/jamcm.2017.1.02
Boulaaras, S., Hocine, M. & Haiour, M., 2017. L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms. Journal of Applied Mathematics and Computational Mechanics, 16(1), pp.19-31. Available at: https://doi.org/10.17512/jamcm.2017.1.02
[1]S. Boulaaras, M. Hocine and M. Haiour, "L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 1, pp. 19-31, 2017.
Boulaaras, Salah, Med Amine Bencheikh Le Hocine, and Mohamed Haiour. "L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms." Journal of Applied Mathematics and Computational Mechanics 16.1 (2017): 19-31. CrossRef. Web.
1. Boulaaras S, Hocine M, Haiour M. L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(1):19-31. Available from: https://doi.org/10.17512/jamcm.2017.1.02
Boulaaras, Salah, Med Amine Bencheikh Le Hocine, and Mohamed Haiour. "L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms." Journal of Applied Mathematics and Computational Mechanics 16, no. 1 (2017): 19-31. doi:10.17512/jamcm.2017.1.02
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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