Appl. Math. Inf. Sci. 9, No. 3, 1247-1257 (2015) 1247 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090318 A General Case for the Maximum Norm Analysis of an Overlapping Schwarz Methods of Evolutionary HJB Equation with Nonlinear Source Terms with the Mixed Boundary Conditions Salah Boulaaras 1,2,∗ and Mohamed Haiour 3 1 Department of Mathematics, Colleague of Science and Arts, Al-Ras, Al-Qassim University, Kingdom Of Saudi Arabia 2 Laboratory of Fundamental and Applied Mathematics, As-Sania University, Oran, Algeria 3 Department of Mathematics, Faculty of Science, University of Annaba, Box. 12, Annaba 23000. Algeria Received: 19 Jul. 2014, Revised: 20 Oct. 2014, Accepted: 21 Oct. 2014 Published online: 1 May 2015 Abstract: In this paper we provide a maximum norm analysis of an overlapping Schwarz method on non-matching grids for evolutionary HJB equation with nonlinear source terms with the mixed boundary conditions and a general elliptic operator. Moreover, an asymptotic behavior in uniform norm is established. Keywords: Domain Decomposition, HJB equation, PQVIs,Error Estimate, Asymptotic Behavior 1 Introduction The main work of this paper is to extend the previous numerical analysis results ([3], [4], [5]) to the following new evolutionary HJB equations with mixed boundary conditions and the general elliptic operator: find u (x, t ) such that u ∈ L 2 (0, T ; K (u)) , u t ∈ L 2 ( 0, T ; L 2 (Ω ) )                  ∂ u i ∂ t + max i=1,...,M ( A i u − f i (u) ) = 0, in Σ , ∂ u i ∂η = ψ i in Γ 0 , i = 1, ..., M, u i = 0 in Γ /Γ 0 , u i (x, 0)= u i 0 in Ω (1) where Ω is a bounded smooth domain in R d , d ≥ 1 and Σ is a set in R × R d defined as Σ =[0, T ] × Ω with T < +∞. A i are the differential operators defined as follows A i = − N ∑ j,k=1 ∂ ∂ x j a i jk (x) ∂ ∂ x k + N ∑ k=1 b i k (x) ∂ ∂ x k + a i 0 (x) (2) and their bilinear forms are associated with A i ; for u, v ∈ H 1 0 (Ω ) a i (u, v)=  Ω  N ∑ j,k=1 a i jk (x) ∂ u ∂ x j ∂ v ∂ x k dx  + +  Ω  N ∑ j=1 b i k (x) ∂ u ∂ x j v + a i 0 (x) uvdx  , (3) assumed to be noncoercive. and the smooth functions a i k, j (x), b i k (x) , a i 0 (x) ∈ ( L ∞ (Ω ) ∩ C 2 ( ¯ Ω )) M , x ∈ ¯ Ω , 1 ≤ k, j ≤ N are sufficiently smooth coefficients and satisfy the following conditions a i jk (x)= a i kj (x); a i 0 (x) ≥ β > 0, β is a constant (4) such that N ∑ j,k=1 a i jk (x)ξ j ξ k ≥ γ |ξ | 2 ; ξ ∈ R N , γ > 0, x ∈ ¯ Ω (5) ∗ Corresponding author e-mail: saleh boulaares@yahoo.fr c  2015 NSP Natural Sciences Publishing Cor.