TY - JOUR

T1 - A divergence-free stabilised finite element method for the evolutionary Navier-Stokes equations

AU - Allendes, Alejandro

AU - Barrenechea, Gabriel R.

AU - Novo, Julia

PY - 2021/6/29

Y1 - 2021/6/29

N2 - 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.

AB - 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.

KW - evolutionary Navier–Stokes equations

KW - stabilised ﬁnite element methods

KW - divergence-free ﬁnite element method

UR - https://www.siam.org/publications/journals/siam-journal-on-scientific-computing-sisc

M3 - Article

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

ER -