  Stabilised mixed velocity-pressure formulations are one of the widely-used
finite element schemes for computing the numerical solutions of laminar
incompressible Navier-Stokes. In these formulations, the Newton-Raphson scheme
is employed to solve the nonlinearity in the convection term. One fundamental
issue with this approach is the computational cost incurred in the
Newton-Raphson iterations at every load/time step. In this paper, we present an
iteration-free mixed finite element formulation for incompressible
Navier-Stokes that preserves second-order temporal accuracy of the
generalised-alpha and related schemes for both velocity and pressure fields.
First, we demonstrate the second-order temporal accuracy using numerical
convergence studies for an example with a manufactured solution. Later, we
assess the accuracy and the computational benefits of the proposed scheme by
studying the benchmark example of flow past a fixed circular cylinder. Towards
showcasing the applicability of the proposed technique in a wider context, the
inf-sup stable P2-P1 pair for the formulation without stabilisation is also
considered. Finally, the resulting benefits of using the proposed scheme for
fluid-structure interaction problems is illustrated using two benchmark
examples in fluid-flexible structure interaction.
