Monolithic Navier–Stokes solver with PETSc field-split (LSC) preconditioning

[akarve1507]

I would like to suggest adding a monolithic incompressible Navier–Stokes solver example to the FEniCSx tutorial collection.
This implementation would solve the fully coupled velocity–pressure system using PETSc field-split (Schur complement) preconditioning, starting with the Least-Squares Commutator (LSC) approach. The example would build upon existing Navier–Stokes tutorials (ns_code1, ns_code2) but differ by demonstrating a block-preconditioned monolithic formulation rather than a projection or splitting scheme.

The tutorial could include standard benchmark problems such as the lid-driven cavity or steady cylinder flow, showcasing performance and solver configuration details for users interested in advanced PETSc options.

Could you please confirm whether this capability already exists in the repository or if this would be a new and valuable addition?

Thank you for your time and consideration.

[jorgensd]

I think we could replace ns_code1 with a block preconditioned, monolithic solver. That demo enforces bad behavior (pressure enforced strongly), and introduces a sub-par splitting scheme (ns_code2.py is better).

so yes, I would be interested in a demo that does this. Maybe open cavity example from: https://arxiv.org/pdf/2112.05309

I am open to other benchmarks that have available data as well.

[akarve1507]

Thank you for the clarification and helpful feedback.

I’ve reviewed the Open Cavity benchmark from Huang et al. (2021, arXiv:2112.05309), and agree it’s an excellent choice.

Based on your suggestion, I plan to:

• Replace the existing ns_code1.py with a monolithic incompressible Navier–Stokes solver using PETSc field-split (LSC) preconditioning.
• Use the Open Cavity configuration: inflow u=(1−e−5t,0)u=(1−e−5t,0), zero-Neumann outflow, and no-slip top and bottom walls, tested for multiple viscosity values.
• Implement the solver using Taylor–Hood (P2–P1) elements with a steady-state Picard/Newton iteration.
• Include velocity and pressure comparisons across viscosity values to reproduce the benchmark’s results.

Before proceeding, I just wanted to confirm that this setup aligns with what you had in mind for replacing ns_code1.py.

Once confirmed, I’ll open a new GitHub Issue to formally outline the implementation plan, and then submit a Pull Request (PR) once the new solver example is complete.

Thank you again for the guidance and I really appreciate your time and input.

[jorgensd]

What about using another inf-sup stable finite pair, for instance a div-conforming pair, CR? (https://jsdokken.com/fenics22-tutorial/comparing_elements.html#crouzeix-raviart) or any other ones ?

[jorgensd]

Other than that, this seems like a good approach

[akarve1507]

Thank you for the suggestion, that’s a great point.

Yes, using another inf-sup stable element pair, such as the Crouzeix–Raviart (CR) element or another div-conforming option, could indeed be valuable.

Would you recommend that I implement the solver directly using the CR element pair instead of the Taylor–Hood (P2–P1) formulation, or should I start with the Taylor–Hood version first and then explore the CR-based implementation as a comparison?

Appreciate the guidance.

[jorgensd]

That is up to you. I would probably go straight for CR.

[akarve1507]

Hi @jorgensd . Thanks for the guidance. I’ll proceed with a CR–P0 formulation (Crouzeix–Raviart velocity, DG-0 pressure) for the monolithic Navier–Stokes example, using a PETSc fieldsplit Schur preconditioner.
For the implementation, I’ll start with a SIMPLE / self-p Schur approximation (block-preconditioned, monolithic) to keep the scope focused, and use the Open Cavity benchmark as discussed. Please let me know if you prefer a specific approach to enforce mean-zero pressure (constraint vs. pinning one DOF).