The finite element approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms

This paper deals with the semi-implicit scheme with respect to the t -variable combined with a finite element spatial approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms. We establish a convergence and a quasi-optimal L ∞ -asymptotic behavior, involving a weakly coupled system of discrete parabolic quasi-variational inequalities (PQVIs), for the solution of which an iterative discrete scheme of monotone kind is introduced and analyzed. Furthermore, the simple numerical example shows that the estimates introduced in this paper are efficient.