Numerical approximation of the Voigt regularization for incompressible Navier–Stokes and magnetohydrodynamic flows

We study Voigt regularizations for the Navier–Stokes equations (NSEs) and magnetohydrodynamic (MHD) equations in the presence of physical boundary conditions. In particular, we develop the first finite element numerical algorithms for these systems, prove stability and convergence of the algorithms, and test them computationally on problems of practical interest. It is found that unconditionally stable implementations of the Voigt regularization can be made from a simple change to existing NSE and MHD codes, and moreover, optimal convergence of the developed algorithms’ solutions to physical solutions can be obtained if lower-order mixed finite elements are used. Finally, we show that for several benchmark problems, the Voigt regularization on a coarse mesh produces good approximations to NSE and MHD systems; that is, the Voigt regularization provides accurate reduced order models for NSE and MHD flows.