Divergence-Conforming Isogeometric Collocation Methods for the Incompressible Navier-Stokes Equations

TL;DR

This paper introduces divergence-conforming isogeometric collocation methods for incompressible Navier-Stokes equations, utilizing B-splines and structure-preserving transformations to improve convergence on complex domains.

Contribution

The paper presents two novel divergence-conforming collocation schemes for incompressible flow, incorporating vorticity and demonstrating enhanced convergence rates over traditional methods.

Findings

The vorticity-velocity-pressure scheme accelerates convergence.

Methods effectively handle complex domain geometries.

Numerical results validate the approach's promise.

Abstract

We develop two isogeometric divergence-conforming collocation schemes for incompressible flow. The first is based on the standard, velocity-pressure formulation of the Navier-Stokes equations, while the second is based on the rotational form and includes the vorticity as an unknown in addition to the velocity and pressure. We describe the process of discretizing each unknown using B-splines that conform to a discrete de Rham complex and collocating each governing equation at the Greville abcissae corresponding to each discrete space. Results on complex domains are obtained by mapping the equations back to a parametric domain using structure-preserving transformations. Numerical results show the promise of the method, including accelerated convergence rates of the three field, vorticity-velocity-pressure scheme when compared to the two field, velocity-pressure scheme.