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

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.

References in corpus (4)

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• Divergence conforming DG method for the optimal control of the Oseen equation with variable viscosity
• Numerical analysis of a new formulation for the Oseen equations in terms of vorticity and Bernoulli pressure