## Trivariate Spline Approximations of 3D Navier-Stokes Equations

**Abstract :**
We present numerical approximations of the 3D steady state Navier-Stokes
equations in velocity-pressure formulation. We use trivariate splines of
arbitrary degree d and arbitrary smoothness r < d. Using functional
arguments, we derive the discrete Navier-Stokes equations in terms of
B-coefficients of trivariate splines over a tetrahedral partition of any
given polygonal domain. Smoothness conditions, boundary conditions and the
divergence-free conditions are enforced through Lagrange multipliers. The
discrete equations are solved by a variant of the augmented Lagrangian
algorithm for which we prove a linear algebraic convergence rate. We have
implemented this approach in MATLAb and present numerical evidence of the
convergence rate as well as experiments on the lid driven cavity flow
problem.