A Mass-Conservative Reduced-Order Algorithm in Solving Optimal Control of Convection-Diffusion Equation

Abstract

This paper introduces a novel approach, the mass-conservative reduced-order characteristic finite element (MCROCFE) method, designed for optimal control problem governed by convection-diffusion equation. The study delves into scenarios where the original state equation exhibits mass-conservation, yet the velocity field is non-divergence-free. The key points of emphasis are: (1) The method effectively addresses convection-dominated diffusion systems through the application of the characteristic technique; (2) Its efficiency is underscored by leveraging the proper orthogonal decomposition (POD) technique, significantly reducing the scale of solving algebraic equation systems; (3) The proposed scheme, based on the mass-conservative characteristic finite element (MCCFE) method framework and the classical POD technique with a slight adjustment to reduce-order space, maintains mass-conservation for the state variable. A priori error estimates are derived for the mass-conservative reduced-order scheme. Theoretical results are validated through numerical comparisons with the MCCFE method, emphasizing the mass-conservation, accuracy and efficiency of the MCROCFE method.