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

Madjda Miloudi,Samira Saadi,Mohamed Haiour

doi:10.1515/dema-2021-0043

Abstract

Abstract 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.

Highlights

We consider the following Hamilton-Jacobi-Bellman (HJB) equations with nonlinear source terms and mixed boundary conditions: find u ∈ W2,∞(Ω), such that:⎧ max (Aiu − f i(u)) = 0 in Ω, ⎪1≤i≤M ∂u ⎨ ⎪ ∂η = φ in Γ0, (1)⎩u = 0 on Γ/Γ0, where Ω is a bounded open set of N, N ≥ 1 with smooth boundary Γ, Γ0 = {x ∈ Γ such that ∀ξ > 0, x + ξ ∉ Ω}, and A1,..., AM denote uniformly second-order elliptic operators defined by
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]

Summary

Introduction

We consider the following Hamilton-Jacobi-Bellman (HJB) equations with nonlinear source terms and mixed boundary conditions: find u ∈ W2,∞(Ω), such that:. We first establish an error estimate between the continuous algorithm and its finite element version and between the exact solution and the finite element approximate We exploit this idea to derive an optimal convergence order for the HJB equation. 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 studied Schwarz methods of parabolic HJB equation with nonlinear source terms with mixed boundary conditions [10]. It is shown in [3] that (1) can be approximated by the following weakly coupled system of quasivariational inequalities (QVIs) with mixed boundary conditions. Passing to the limit, as k → 0, we get ∥Tω − Tω ∥∞ ≤ max ∥ξ i − ξ∼i∥∞ ≤ ρ ∥ω − ω ∥∞ , T is a contraction

The discrete problem

A continuous iterative scheme

A discrete iterative scheme

Conclusion