Analysis and approximation of a vorticity-velocity-pressure formulation for the Oseen equations

TL;DR

This paper develops and analyzes mixed finite element and discontinuous Galerkin methods for the Oseen equations, ensuring divergence-free velocities and optimal convergence, supported by theoretical proofs and numerical tests.

Contribution

It introduces new discretization schemes for the Oseen equations with provable stability and convergence, including divergence-free velocity approximation and stabilization techniques.

Findings

Finite element schemes are well-defined with optimal convergence.

Discontinuous Galerkin schemes achieve optimal convergence with stabilization.

Numerical examples confirm theoretical properties.

Abstract

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.