Replace ns_code1.py with a monolithic Navier–Stokes example (CR–P0) using PETSc field-split Schur preconditioning

[akarve1507]

Description
This issue proposes replacing the current ns_code1.py example with a monolithic incompressible Navier–Stokes solver implemented using the Crouzeix–Raviart (CR) element for velocity and DG-0 for pressure (CR–P0 pair).
The solver will employ PETSc’s field-split (Schur complement) preconditioning to demonstrate a modern, scalable monolithic formulation for incompressible flow problems in DOLFINx.
This update will align the Navier–Stokes tutorials with current PETSc practices and provide a clearer, physically accurate alternative to the existing projection/splitting scheme.

Objectives

• Implement a steady-state monolithic Navier–Stokes solver using Picard iteration.
• Use CR–P0 elements for an inf-sup stable, div-conforming discretization.
• Apply PETSc field-split Schur preconditioning with the selfp (SIMPLE-type) Schur approximation.
• Replace ns_code1.py with this new example and corresponding notebook.
• Benchmark the solver using the Open Cavity case from Huang et al., “A Large-Scale Benchmark for the Incompressible Navier–Stokes Equations” (arXiv:2112.05309).

Benchmark Setup

• Domain: 2D open cavity (rectangular domain).
• Inflow: u=(1−e −5t,0) on the left boundary.
• Outflow: Zero-Neumann (traction-free).
• Walls: No-slip top and bottom.
• Viscosity range: ν=0.0005–0.01.
• Pressure constraint: Mean-zero enforced by pinning one pressure DOF.

[jorgensd]

Minor comments, for the sake of searchability, I’ll keep the ns1 code online, but will mark it as outdated.

Why do you need zero mean pressure if there is an outlet with Neumann?

[akarve1507]

Thank you, that makes sense and thank for pointing that out.
I initially included the zero-mean pressure condition as a safeguard against potential singularities, since in fully Dirichlet (closed cavity) setups, the pressure is only determined up to a constant. However, in this Open Cavity case, the Neumann outlet already provides a natural pressure reference through the traction-free condition, so the pressure field should be uniquely defined without enforcing mean-zero.
I’ll proceed without the explicit pressure constraint and let the outlet boundary condition anchor the pressure level as intended.