Analysis and approximation of a vorticity-velocity-pressure formulation for the Oseen equations
This work provides new numerical schemes for fluid dynamics problems, but the contribution is incremental as it extends existing formulations to a specific set of equations.
The paper introduces mixed and discontinuous Galerkin methods for solving the Oseen equations in vorticity-velocity-pressure form, proving well-posedness and optimal convergence rates with exactly divergence-free velocities.
We introduce a family of mixed methods and discontinuous Galerkin discretisations designed to numerically solve the Oseen equations written in terms of velocity, vorticity, and Bernoulli pressure. The unique solvability of the continuous problem is addressed by invoking a global inf-sup property in an adequate abstract setting for non-symmetric systems. The proposed finite element schemes, which produce exactly divergence-free discrete velocities, are shown to be well-defined and optimal convergence rates are derived in suitable norms. In addition, we establish optimal rates of convergence for a class of discontinuous Galerkin schemes, which employ stabilisation. A set of numerical examples serves to illustrate salient features of these methods.