Higher order spectral/hp finite element models of the Navier–Stokes equations

Abstract

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.

Keywords:

• Navier–Stokes equations
• least-squares formulations
• first-order formulations
• higher order finite elements
• spectral/hp methods

Acknowledgements

The authors would like to express their sincere gratitude to the anonymous reviewers for their constructive comments during the review process.

Additional information

Funding

The research reported herein was carried out with the support of Qatar National Research Fund, under the grant of National Priorities Research Program [grant number NPRP 09-1183-2-461]. The authors gratefully acknowledge the support of Supercomputing Facility at Texas A&M University, College Station.