Conforming/Non-Conforming Mixed Finite Element Methods for Optimal Control of Velocity-Vorticity-Pressure Formulation for the Oseen Problem with Variable Viscosity

TL;DR

This paper introduces new conforming and non-conforming mixed finite element methods for optimal control of the velocity-vorticity-pressure formulation of the generalized Oseen problem with variable viscosity, providing theoretical analysis and numerical validation.

Contribution

It proposes and analyzes novel conforming augmented mixed finite element and DG methods for the velocity-vorticity-pressure formulation, including error estimates and numerical experiments.

Findings

Optimal a priori error estimates established

Residual-based a posteriori error estimates derived

Numerical experiments demonstrate method effectiveness

Abstract

This work examines the distributed optimal control of generalized Oseen equations with non-constant viscosity. We propose and analyze a new conforming augmented mixed finite element method and a Discontinuous Galerkin (DG) method for the velocity-vorticity-pressure formulation. The continuous formulation, which incorporates least-squares terms from both the constitutive equation and the incompressibility condition, is well-posed under certain assumptions on the viscosity parameter. The CG method is divergence-conforming and suits any Stokes inf-sup stable velocity-pressure finite element pair, while a generic discrete space approximates vorticity. The DG scheme employs a stabilization technique, and a piecewise constant discretization estimates the control variable. We establish optimal a priori and residual-based a posteriori error estimates for the proposed schemes. Finally, we provide numerical experiments to showcase the method's performance and effectiveness.