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The finite element approximation of evolutionary Hamilton-Jacobi-Bellman equations with nonlinear source terms. (English) Zbl 1254.65104
Summary: 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 $$L^{\infty }$$-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.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 49J40 Variational inequalities 35Q53 KdV equations (Korteweg-de Vries equations) 35K99 Parabolic equations and parabolic systems
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