This work is devoted to the ﬁnite element discretisation of the incompressible Navier-Stokes equations. The starting point is a low-order stabilised ﬁnite element method using piecewise linear continuous discrete velocities and piecewise constant pressures. This pair of spaces needs to be stabilised, and, as such, the continuity equation is modiﬁed by adding a stabilising bilinear form based on the jumps of the pressure. This modiﬁed continuity equation can be rewritten in a standard way involving a modiﬁed diﬀerent velocity ﬁeld, which is as a consequence divergence-free. This modiﬁed velocity ﬁeld is then fed back to the momentum equation making the convective term skew-symmetric. Thus, the discrete problem can be proven stable without the need to rewrite the convective ﬁeld in its skew-symmetric way. Error estimates with constant independent of the viscosity are proven. Numerous numerical experiments conﬁrm the theoretical results.
|Journal||SIAM Journal on Scientific Computing|
|Publication status||Accepted/In press - 29 Jun 2021|
- evolutionary Navier–Stokes equations
- stabilised ﬁnite element methods
- divergence-free ﬁnite element method