Conforming/Non-Conforming Mixed Finite Element Methods for Optimal Control of Velocity-Vorticity-Pressure Formulation for the Oseen Problem with Variable Viscosity
<|reference_start|>Conforming/Non-Conforming Mixed Finite Element Methods for Optimal Control of Velocity-Vorticity-Pressure Formulation for the Oseen Problem with Variable Viscosity: 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.<|reference_end|>
@article{singh2024conforming/non-conforming,
    title={Conforming/Non-Conforming Mixed Finite Element Methods for Optimal
  Control of Velocity-Vorticity-Pressure Formulation for the Oseen Problem with
  Variable Viscosity},
    author={Harpal Singh, Arbaz Khan},
    journal={arXiv preprint arXiv:2407.01977},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.01977},
    primaryClass={math.NA cs.NA}
}