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The main purpose of this paper is to analyze the convergence of the proposed algorithm 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: 65N30

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