The Theta Time Scheme Combined with a Finite-Element Spatial Approximation in the Evolutionary Hamilton–Jacobi–Bellman Equation with Linear Source Terms

Abstract

This paper is an extension and generalization of the previous results, cf. [2---4]. It is devoted to the theta-scheme with respect to the t -variable combined with a finite-element spatial approximation of the evolutionary Hamilton---Jacobi---Bellman equations (HJB equation) and involves a weakly coupled discrete system of parabolic quasi-variational inequalities (PQVs). Its relation to time energy behavior is proved. In addition, the PQVs are transformed into a coercive discrete system of elliptic quasi-variational inequalities. A new iterative discrete algorithm is also proposed to show the existence and uniqueness of the discrete solution. Moreover, its convergence is established. Then a simple proof to an asymptotic behavior in uniform norm is given. Furthermore the proposed approach is based on a discrete L ? -stability property with respect to the right-hand side and the boundary conditions.