The nonconforming virtual element method for optimal control problem governed by Stokes equations

In this paper we investigate the lowest-order nonconforming virtual element approximation of optimal control problem governed by Stokes equations. Based on the lowest-order virtual element approximation of the state equation and variational discretization of the control variable, we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2$$\end{document} and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>1$$\end{document} norms are derived. Moreover, the presented method can also handle general polygonal meshes with arbitrary nodes (including non-convex and degenerate elements). Numerical experiments are carried out to confirm the convergence analysis and illustrate the theoretical findings.