L
 ∞-error estimates of a finite element method for Hamilton-Jacobi-Bellman equations with nonlinear source terms with mixed boundary condition


 In this paper, we introduce a new method to analyze the convergence of the standard finite element method for Hamilton-Jacobi-Bellman equation with noncoercive operators with nonlinear source terms with the mixed boundary conditions. The method consists of combining Bensoussan-Lions algorithm with the characterization of the solution, in both the continuous and discrete contexts, as fixed point of contraction. Optimal error estimates are then derived, first between the continuous algorithm and its finite element counterpart and then between the continuous solution and the approximate solution.


Introduction
We consider the following Hamilton-Jacobi-Bellman (HJB) equations with nonlinear source terms and mixed boundary conditions: find ( ) ∈ ∞ u W Ω 2, , such that:

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We also define the associated bilinear forms We specify the following notations: ∥ ∥ ∥∥ ∥∥ ∥∥ . , . . In this paper, we are concerned with the numerical approximation in the ∞ L norm for the problem (1), and we instead combine, in both the continuous and discrete contexts, the Bensoussan-Lions algorithm with the characterization of the solution as a fixed point of a contraction. We first establish an error estimate between the continuous algorithm and its finite element version and then between the exact solution and the finite element approximate. We exploit this idea to derive an optimal convergence order for the HJB equation.
This method consists, mainly, of combining, in both the continuous and discrete contexts, the concept of fixed point and a geometrical convergence of an iterative scheme approximating the solution. For a computational purpose, this method provides an interesting information as it permits to control the error between the continuous iterative scheme and its finite element counterpart.
While in previous studies, Cortey Dumont and Boulbrachene used the concept of subsolutions to find the estimation of error.
The HJB equation has been analytically studied in [1][2][3][4]. For the numerical approximations, Cortey Dumont [5] investigated a finite element approximation and used a subsolution method. Boulbrachene and Haiour [6] investigated a finite element Bensoussan-Lions algorithm version and obtained a quasi-optimal error estimate in the ∞ L -norm. Boulbrachene and Cortey Dumont [7] investigated a finite element method using the concept of subsolutions and discrete regularity and obtained an optimal error estimate in the ∞ L -norm. Boulaaras and Haiour investigated a finite element of the HJB equation elliptic and parabolic [8,9]. They also studied Schwarz methods of parabolic HJB equation with nonlinear source terms with mixed boundary conditions [10].
This paper is organized as follows: We view the continuous problem in Section 1 and the discrete problem in Section 2. We address the continuous algorithm in Section 3 and the discrete algorithm in Section 4, and we establish, in both the continuous and discrete cases, the geometrical convergence of this algorithms. Finally, in Section 5, we present the finite element error analysis.
It is shown in [3] that (1) can be approximated by the following weakly coupled system of quasivariational inequalities (QVIs) with mixed boundary conditions where k is a positive constant. This is precisely stated in the following theorem.
The system (3) has a unique solution, which belongs to . Moreover, as → k 0, each component of this system converges uniformly in ( ) C Ω to the solution u of HJB equation (1).

Lemma 2. [11]
There exists a constant c independent of k, thus where ξ is the unique solution of the following HJB equation: From [3], (4) can be approximated by the following system of QVIs where k is a positive constant and we have ∥ There exists a constant c independent of k, thus be the corresponding solutions of the following system of quasi-variational inequalities: , . Then: Thus, making use of monotonicity result with respect to right-hand side for system of QVIs related to HJB equation with boundary mixed conditions (see [5]), we get: Hence, passing to the limit, as → k 0, we get Thus, T is a contraction. □

The discrete problem
Let Ω be decomposed into triangles, τ h denote the set of all those elements, and > h 0 be the mesh size. We assume that the family τ h is regular and quasi-uniform. Let be the finite element space, where K is a triangle of τ h and P 1 is the space of polynomials with degree ≤1.
be the basis functions of the space V h , and A i the matrices with generic coefficients i ls i l s Let us also define the discrete right-hand sides  It was shown in [5] that (6) can be approximated by the following discrete weakly coupled system of QVIs Theorem 9.
[5] Under condition of Lemma 3, then, the system (7) has a unique solution. Moreover, as → k 0; each component of the solution of this system converges uniformly in ( ) C Ω to the solution u h of (6).
We introduce the mapping where ξ h is the unique solution of the following discrete coercive HJB equation the discrete coercive HJB equation (8) can be approximated by the following system of QVIs Proof. Exactly the same as that of Lemma 2. □ Theorem 11. Under condition of Lemma 3, the mapping T h is a contraction, so the solution of discrete HJB equation (6) is its unique fixed point.
Proof. Exactly the same as that of Theorem 2. □

A continuous iterative scheme
is the unique solution of the variational equation: be the unique solution of the system of QVIs, which approximated the coercive HJB equation (9): such that each iterate u h n solves the discrete coercive HJB equation: be the unique solution of the system of QVIs, which approximated the discrete coercive HJB equation (11):   be the unique solution of the system of QVIs, which approximated the discrete coercive HJB equation (13)