A mass-conservative characteristic FE scheme for optimal control problems governed by convection–diffusion equations

Abstract

In this paper, we develop a mass-conservative characteristic finite element scheme for optimal control problems governed by linear singularly perturbed, convection–diffusion equations. We discuss the case that the velocity fields are non-divergence-free. The space discretization of the state variable is done by piecewise linear continuous functions, whereas the control variable is approximated by piecewise constant functions due to the limitation of the regularity. The scheme preserves the mass balance for the original state equation. We derive a priori error estimates for both the control and state approximations. Some numerical examples are presented to show the efficiency of the proposed scheme.